Sports Leagues Scheduling: Models, Combinatorial Properties, and Optimization Algorithms (Lecture Notes in Economics and Mathematical Systems)

Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr...

Author: Dirk Briskorn

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Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten

603

Dirk Briskorn

Sports Leagues Scheduling Models, Combinatorial Properties, and Optimization Algorithms

123

Dirk Briskorn Department of Production and Logistics University of Kiel Olshausenstrasse 40 24098 Kiel Germany [email protected]

ISBN 978-3-540-75517-3

e-ISBN 978-3-540-75518-0

DOI 10.1007/978-3-540-75518-0 Lecture Notes in Economics and Mathematical Systems ISSN 0075-8442 Library of Congress Control Number: 2007938899 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com

Basic research is like shooting an arrow into the air and, where it lands, painting a target. Homer Adkins

Preface

This book is the result of my research on sports leagues scheduling at the Christian-Albrechts-University of Kiel. This research has been done during my employment as research associate at the Chair for Production and Logistics. The challenging research topic as well as the friendly environment made these years enjoyable and pleasant. First of all, I wish to express my gratitude to my thesis advisor, Professor Dr. Andreas Drexl. He has established a very inspiring and motivating research environment, and he always took the time for discussions and helpful advices. Moreover, he refereed this work. Furthermore, I am very grateful to Professor Dr. S¨ onke Albers for co-refereeing this thesis. Additionally, I would like to thank my colleagues in Kiel for many helpful discussions, comments, and suggestions. I am especially grateful to Dr. Andrei Horbach, Marcel B¨ uther, and Dr. Yury Nikulin. Stefan Wende was a big help regarding the technological background of my work. Ethel Fritz, Jens Heckmann, and J¨ urgen Lux were also always helpful with most various things. Besides, I thank Professor Frits Spieksma, Juniorprofessorin Dr. Sigrid Knust, and Dr. Thomas Bartsch for stimulating discussions and advice. Finally, I would like to thank my family and, last but not least, Eva for supporting me and bearing me.

Kiel, December 2007

Dirk Briskorn

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4

2

Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Single Round Robin Tournament . . . . . . . . . . . . . . . . . . . . . 2.2 Double Round Robin Tournament . . . . . . . . . . . . . . . . . . . . 2.3 r Round Robin Tournament . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Decomposition Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 First–Schedule–Then–Break . . . . . . . . . . . . . . . . . . . . 2.4.2 First–Break–Then–Schedule . . . . . . . . . . . . . . . . . . . .

5 6 12 16 18 19 22

3

Real World Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Externally Given Constraints . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Forbidden Matches . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Regions’ Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Highly Attended Matches . . . . . . . . . . . . . . . . . . . . . . 3.2 Fairness Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Opponents’ Strengths . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Teams’ Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Computational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Generating Problem Instances . . . . . . . . . . . . . . . . . . 3.3.2 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 30 31 32 33 33 34 40 42 42 43 57

X

Contents

4

Combinatorial Properties of Strength Groups . . . . . . . . 4.1 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Ordered 1-Factorization of Kk,k . . . . . . . . . . . . . . . . 4.1.2 Ordered 1-Factorizations of Kk . . . . . . . . . . . . . . . . . 4.1.3 Ordered Symmetric 2-Factorization of 2K2k+1 . . . . 4.2 Group-Balanced Single Round Robin Tournaments . . . . . 4.3 Group-Changing Single Round Robin Tournaments . . . . . 4.4 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 60 60 64 67 70 75 77

5

Home-Away-Pattern Based Branching Schemes . . . . . . 79 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 General Home-Away-Pattern Sets . . . . . . . . . . . . . . . . . . . . 80 5.2.1 Achieving Feasible Home-Away-Pattern Sets . . . . . 81 5.2.2 Choice of Branching Candidates . . . . . . . . . . . . . . . . 83 5.2.3 Node Order Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 Minimum Number of Breaks . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3.1 Achieving Feasible Home-Away-Pattern Sets . . . . . 87 5.3.2 Choice of Branching Candidates . . . . . . . . . . . . . . . . 95 5.3.3 Node Order Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4.1 General Home-Away-Pattern Sets . . . . . . . . . . . . . . . 98 5.4.2 Minimum Number of Breaks . . . . . . . . . . . . . . . . . . . 99 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6

Branch–and–Price Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1 Motivation and Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.1 Set Partitioning Master Problem . . . . . . . . . . . . . . . 104 6.2.2 Matching Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 Branching Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3.1 Branching Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3.2 Node Order Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4 Column Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4.1 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4.2 Column Management . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4.3 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.5 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.6 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7

Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Contents

XI

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 List of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 List of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

1 Introduction

1.1 Motivation Sports league scheduling is an essential activity arising in the context of sports events organization. Sports events are of great importance as far as economic aspects are considered. Often, a sports club is a major employer and taxpayer. Thus, private persons as well as public agencies depend on particular sports events as well as on regular sports league seasons. In this research the focus is on regular sports seasons. A sports league schedule (SLS) determines the date and the venue of a match between two opponents. Scheduling must be done by a central agent since each club has specific interests affecting each other. Therefore, a sports league can be seen as a “supplier” of a season. The “customers” are composed of fans watching matches in the stadium and tv channels broadcasting them either live or retarded as a summary. Furthermore, sports clubs make money by offering food supply, selling merchandize assortments such as caps, shirts, scarfs, and so on. While the latter highly depends on the current success of a specific club, the former can be supported by constructing pleasant SLSs. Of course, along with attractiveness there are a lot of attributes with respect to the schedules’ structure, security aspects, resources, and infrastructure to be considered. There are few standard construction schemes for SLSs which all fail if real world constraints are taken into account. It is no surprise that schedules even for professional sports leagues are constructed manually due to the lack of adequate planning tools as reported in Bartsch [5]. Recent research activities have led to several promising approaches, see for example Bartsch et al. [6] and Nemhauser and Trick [68]. However,

2

1 Introduction

due to the tremendous number of possible SLSs and the sheer difficulty to find even one nearly all existing approaches inspect a rather small part of the solution space. Hence, many potential SLSs are cut out. Another popular approach is to search only one feasible solution neglecting better ones. Thus, there is a great need for efficient scheduling approaches. In this field tackling the whole solution space might be an especially challenging task. Additionally, heuristics finding good solutions in an acceptable amount of CPU time are of practical interest.

1.2 Related Work There is a vast field of literature concerning sports league scheduling and related topics. In the following we present an overview. There are several common modelling ideas. For example a popular analogy between SLSs and an edge coloring of complete graphs does exist. Consequently, many articles dealing with graph-based models can be found, for example see de Werra [19, 20, 21, 22, 23], de Werra et al. [24], and Drexl and Knust [30]. Brucker and Knust [13] and Drexl and Knust [30] deal with sports league scheduling problems formulated as multi-mode resource constrained project scheduling problems. Note that an edge coloring of a complete graph Kn with n − 1 colors is equivalent to a 1-factorization of the same graph being closely related to a latin square of size n. A couple of papers study these analogies from a design theory point of view, e.g., Rosa and Wallis [71], Gelling and Odeh [43], Easton et al. [34], and Mendelsohn and Rosa [61]. Several articles concern particular integer programming (IP) formulations, e.g., Bartsch et al. [6], Della Croce and Oliveri [25], and Schreuder [76, 77]. In real world sports leagues schedules with different structures appear. Probably the most popular form is a round robin tournament (RRT). In particular, RRTs attract attention as single RRT (see Bartsch et al. [6], Trick [82], and Easton et al. [33]) and as double RRT. Furthermore, research has been done concerning divisions, e.g., in de Werra [22], and multiple venues not related to the teams, e.g., in de Werra et al. [24]. Most researchers focus on constructing a feasible SLS. Nevertheless, there are approaches evaluating different SLSs and trying to find one having the best evaluation. Among the most popular goals are minimizing the number of breaks, e.g., in Elf et al. [36], Fronˇcek and Meszka [41], and Miyashiro and Matsui [63], minimizing travel costs, see Anag-

1.3 Basic Notation

3

nostopoulos et al. [2] and Easton et al. [32], and minimizing carry-overs, see Russell [73]. Many kinds of different sports are considered such as baseball in Russell and Leung [74], basketball in Nemhauser and Trick [68], ice hockey in Fleurent and Ferland [38], soccer in Bartsch et al. [6], de Werra [23] and Schreuder [76], and tennis in Della Croce et al. [26]. Extensive overviews of literature on sports leagues scheduling in the context of operations research are provided by Knust [51] and Rasmussen and Trick [70]. Furthermore, literature can be found on competing strategies for teams, see Machol et al. [58], Gerchak [44] and Ladany and Machol [53] and the motivation for tackling strategy decisions using operations research methods in Mottley [66] and Schutz [79]. Finally, we refer the reader to some articles concerning the economic importance of professional sports for cities or regions in order to emphasize the practical relevance of generating attractive SLSs. Cairns et al. [16], Jeanrenaud [48] as well as Leeds and von Allmen [55] provide extensive surveys taking into account, for example, demand via paid attendances and broadcasts. Furthermore, the relation between sports and economic development is outlined in Baade [3] and Burgan and Mules [14], for example.

1.3 Basic Notation A sports league is a composition of a set T of n, n even, teams competing each other. Competitions can be specified as exactly one team i playing against exactly one other team j and are called matches in the remainder of this work. A match is carried out in exactly one period p out of the set P . A match takes place at one of the both opponents’ stadiums. Therefore, we can identify a match by a triple (i, j, p). Here, p is the period where teams i and j compete at i’s home. As far as not stated otherwise all indices are “1-based”. A SLS in general is a timetable determining the time and the venue where a specific match is carried out. Throughout this work, we consider leagues with an even number of teams obeying a RRT structure. The number of teams being even is no restriction to generality since we can add a dummy team if n is odd and, therefore, unconditionally obtain n being even. The matches of RRTs are grouped in a fashion that each team plays exactly once per period. The collection of all matches carried out in a period is called matchday (MD). Hence, a MD consists of n2 matches. There are several

4

1 Introduction

different RRT structures but they all have in common that each team i plays against each other team j exactly r times with r ∈ N>0 . Then, each SLS contains r(n − 1) MDs and, consequently, |P | = r(n − 1).

1.4 Outline The work is organized as follows. Chapter 2 is focused on basic problems in sports league scheduling. Different structures for RRTs are presented and corresponding optimization problems are introduced. Furthermore, we provide proofs of complexity. In chapter 3 real world requirements are examined. Moreover, we represent them by means of IP model formulations and provide a computational study. We give detailed insights into consideration of strength groups from a combinatorial point of view in chapter 4. In chapter 5 branching schemes based on a well known decomposition scheme are developed. Furthermore, we provide computational results obtained when employing these branching schemes. Chapter 6 provides an highly flexible exact algorithm to be easily adapted for solving all variations of the problem from chapter 3. Finally, chapter 7 gives conclusions and an outlook on future research. All computational results are obtained using a 3.8 GHz Pentium 4 machine with 3 GBs of RAM. We employ Ilog Cplex 9.0 with default settings (if not mentioned otherwise) as standard solver. Run times are given in seconds.

2 Basic Problems

The chapter at hand concerns basic problems solely derived from SLSs’ structural requirements. To this end, we neglect most real world requests in order to focus on complexity induced by the RRT structure itself. Furthermore, we inspect subproblems resulting from popular decomposition schemes. Correspondingly, we define cost minimization problems and give proofs or conjectures for their complexity, respectively. First, we present the well known planar three index assignment problem (PTIAP) which will serve to proof the scheduling problems’ complexity. Definition 2.1. Given are three sets A, B, C with |A| = |B| = |C| = m, m ∈ N, m even, as well as costs da,b,c for each triple (a, b, c) ∈ A × B × C. Feasible solutions to the PTIAP consist of m2 triples such that each pair in (A × B) ∪ (A × C) ∪ (B × C) is contained exactly once. The PTIAP is to find a solution having the minimum sum of chosen triples’ costs. We give here a formulation of PTIAP as an integer program according to, e.g., Spieksma [80] using m3 binary variables and 3m2 constraints. In this formulation ya,b,c equals 1 if triple (a, b, c) is chosen, 0 otherwise. The objective function (2.1) represents the goal of cost minimization. Equations (2.2), (2.3), and (2.4) force each pair to be contained in exactly one chosen triple. In the following we give the decision version of PTIAP which will be referred to as PTIAP-DEC. Definition 2.2. PTIAP-DEC is defined by input and question: Input: Three m-sets A, B, C, and a set D ⊆ A × B × C. Question: Does there exist a subset of D containing m2 triples such

6

2 Basic Problems

Model 2.1: PTIAP-IP min

da,b,c ya,b,c

(2.1)

a∈A b∈B c∈C

s.t.

ya,b,c

=

1

∀ a ∈ A, b ∈ B

(2.2)

ya,b,c

=

1

∀ a ∈ A, c ∈ C

(2.3)

ya,b,c

=

1

∀ b ∈ B, c ∈ C

(2.4)

∈

{0, 1} ∀ a ∈ A, b ∈ B, c ∈ C

(2.5)

c∈C

b∈B

a∈A

ya,b,c

that each pair (a, b) ∈ (A × B), (a, c) ∈ (A × C), and (b, c) ∈ (B × C) is contained exactly once in those triples? In Frieze [40] PTIAP-DEC is proven to be NP-complete implying that PTIAP is NP-hard. Complexity of several sports league scheduling problems will be shown by reduction from PTIAP-DEC hereafter.

2.1 Single Round Robin Tournament Single RRTs obey the general structural requests outlined in section 1.3. Additionally, r is specified to be equal to 1 which means that each team meets each other team exactly once. The tournament contains n − 1 MDs; see table 2.1 for an instance with n = 6. Here i-j denotes team i playing at home against team j. Table 2.1. Single RRT for n = 6 period

1

match 1 1-2 match 2 5-3 match 3 4-6

2

3

4

5

5-6 1-4 2-3

3-4 2-5 1-6

4-5 3-1 2-6

5-1 4-2 3-6

Next, we define the single RRT problem.

2.1 Single Round Robin Tournament

7

Definition 2.3. Given a set T , |T | = n, and a set P , |P | = n − 1, each triple (i, j, p) ∈ T × T × P, i = j, represents a match of team i against team j at i’s home in period p. Costs ci,j,p are given for each match. A feasible solution to the single RRT problem corresponds to a set of n(n−1) triples such that (i) for each pair (i, j) ∈ T × T, i < j, exactly 2 one triple of form (i, j, p) or (j, i, p) with p ∈ P is chosen and such that (ii) for each pair (i, p) ∈ T × P exactly one triple of form (i, j, p) or (j, i, p) with j ∈ T \{i} is chosen. The problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Condition (i) implies that each pair of teams meets exactly once while condition (ii) ensures that each team plays exactly once per period resulting in n − 1 periods. These requests as well as the goal of cost minimization can be represented as an IP model employing n(n − 1)2 constraints. binary variables and 3n(n−1) 2

Model 2.2: SRRTP-IP min

ci,j,p xi,j,p

(2.6)

i∈T j∈T \{i} p∈P

s.t.

(xi,j,p + xj,i,p )

=

1

∀ i, j ∈ T, i < j

(2.7)

=

1

∀ i ∈ T, p ∈ P

(2.8)

∈

{0, 1} ∀ i, j ∈ T, i = j, p ∈ P

(2.9)

p∈P

(xi,j,p + xj,i,p )

j∈T \{i}

xi,j,p

Binary variable xi,j,p is equal to 1 if match (i, j, p) is carried out, and 0 otherwise. Constraints (2.7) and (2.8) correspond to (i) and (ii), respectively, while (2.6) represents the goal of cost minimization. Cost ci,j,p of a specific match can be seen in a rather abstract way here. For example SRRTP-IP can serve as subproblem of a sports scheduling problem taking into account the real world constraints neglected in SRRTP-IP. Then, ci,j,p might cover, among other components, dual variables. However, there are several further applications of SRRTP-IP having practical relevance:

8

2 Basic Problems

• Teams usually have preferences for playing at home in certain periods, a fact which can easily be expressed through ci,j,p . Let pri,p ∈ R be team i’s preference to play at home (pri,p > 0) or to play away (pri,p < 0), respectively, in period p. A preference pri,p is stronger than a preference pri′ ,p′ if |pri,p | > |pri′ ,p′ |. Then, costs can be defined as ci,j,p = −pri,p + prj,p, for example. Here, cost ci,j,p represents neglected preferences of i and j in p decreased by fulfilled ones if (i, j, p) is carried out. Hence, the objective of SRRTP-IP is to maximize the difference between fulfilled preferences and neglected preferences. • Since a major objective of the organizers of a tournament is to maximize attendance we can represent the economic value of the estimated attendance by ci,j,p. Let estimated attendances eai,j,p be given for each match (i, j, p). Then, we can define costs of SRRTPIP as ci,j,p = −eai,j,p and obtain the objective to maximize total tournament’s attendance. Equivalently, ci,j,p can be defined as the number of seats remaining empty in the stadium of i if (i, j, p) is carried out. • Often, a stadium is owned by some public agency and teams have to pay a fee for each match taking place in that particular stadium. This fee might depend on season, day of the week, and time when the match takes place as well as competing events. We can represent it by ci,j,p and obtain the objective to minimize the sum of fees to be paid. • A special case of SRRTP-IP arises when the costs are restricted to {0, 1}. Then ci,j,p = 1 denotes that team i cannot play team j in team i’s home venue in period p, whereas ci,j,p = 0 denotes that this is possible. A reason for a match (i, j, p) being impossible might be restricted availability of team i’s stadium. What we are interested in is to determine whether a feasible schedule, that is, a zero-cost schedule, exists or not. The complexity of the single RRT problem has been independently stated in Briskorn et al. [12] and Easton [31]. The proof in Briskorn et al. [12] is reproduced below for the sake of completeness. In analogy to PTIAP-DEC we define SRRTP-DEC as the decision version of the single RRT problem first. Definition 2.4. SRRTP-DEC is defined by input and question: Input: An instance of single RRT problem having ci,j,p ∈ {0, 1} for each i, j ∈ T , i = j, p ∈ P . Question: Does there exist a solution having cost equal to 0?

2.1 Single Round Robin Tournament

9

Theorem 2.1. The single RRT problem is NP-hard. Proof. We prove theorem 2.1 by presenting a reduction from PTIAPDEC to the single RRT problem. PTIAP-DEC is proven to be NPcomplete in Frieze [40]. First, we reduce PTIAP-DEC to SRRTP-DEC. We assume, without loss of generality, that m is even. Given an instance of PTIAP-DEC, we now build the instance of SRRT-DEC as follows. There are 2m teams, so we have |T | = n = 2m (and of course |P | = 2m − 1). Further, we set 0 ∀ (i, j, p) ∈ D, ci,m+j,p = 1 ∀ (i, j, p) ∈ / D, and

ci,j,p

 1 if i, j, p ∈ {1, . . . , m} , i = j,       1 if i, j ∈ {m + 1, . . . , 2m} , i = j, p ∈ {1, . . . , m} ,      1 if i ∈ {m + 1, . . . , 2m} , j, p ∈ {1, . . . , m} ,       1 if i ∈ {1, . . . , m} , j ∈ {m + 1, . . . , 2m} , p ∈ {m + 1, . . . , 2m − 1} , =     0 if i, j ∈ {1, . . . , m} , i = j, p ∈ {m + 1, . . . , 2m − 1} ,      0 if i, j ∈ {m + 1, . . . , 2m} , i = j, p ∈ {m + 1, . . . , 2m − 1} ,      1 if i ∈ {m + 1, . . . , 2m} , j ∈ {1, . . . , m} ,    p ∈ {m + 1, . . . , 2m − 1} .

This completes the description of the instance of SRRTP-DEC. A yes-answer to the PTIAP-DEC instance corresponds to a yesanswer to the SRRTP-DEC. First, the triples (a, b, c) which constitute the solution of PTIAP-DEC give rise to the following partial solution of SRRTP-DEC: team i = a plays team j = m + b in period p = c at team i’s home venue. Since in this way we use only triples from D, we have ensured that each match between a team i with i ∈ {1, . . . , m}, and a team j with j ∈ {m + 1, . . . , 2m} is scheduled with zero cost. Second, to schedule the remaining matches, let us first deal with the matches between teams i and j with i, j ∈ {1, . . . , m}, i = j. Observe that we must assign these matches to periods m + 1, . . . , 2m − 1 in order to have a zero-cost solution. Assigning these matches to m − 1 periods can be seen as edge-coloring a complete graph (recall that an edge-coloring of a graph is a coloring of the edges such that adjacent edges have different colors). Indeed, the graph that results when there is a vertex for each of the first m teams, and an edge for each match to be played is complete.

10

2 Basic Problems

It is well known (see Mendelsohn and Rosa [61]) that, in case m is even – as we assumed – (m−1) colors suffice to edge color Km . The resulting coloring gives us a feasible assignment of matches to periods (edges with the same color correspond to matches played in the same period). In this way each period receives m 2 matches, each with zero cost. By using the same procedure for teams i and j with i, j ∈ {m + 1, . . . , 2m}, i = j, we find an assignment of the corresponding matches to periods m + 1, . . . , 2m − 1. Hence, we have found a feasible solution to single RRT problem having total cost equal to zero and, therefore, the answer to SRRTP-DEC is yes. Next, if a zero-cost solution to the single RRT problem exists (which means a yes-answer to SRRTP-DEC), it is not difficult to show that PTIAP admits a zero-cost solution, as well. Indeed, let us focus on the matches between teams i and j with i ∈ {1, . . . , m} and j ∈ {m + 1, . . . , 2m}. From the construction it is clear that the existence of a zero-cost solution implies that team j never plays at its home venue against team i since this costs 1. Hence, the assignment of matches of team i against team j to periods p, p = 1, . . . , m (which must exist since we assumed that a zero-cost solution to the single RRT problem exists), gives us the solution to PTIAP and, hence, the answer to PTIAP-DEC is yes. Secondly, we reduce SRRTP-DEC to the single RRT problem. This is straightforward, since we can give an answer to SRRT-DEC by solving an instance of the single RRT problem having costs ci,j,p ∈ {0, 1} corresponding to the costs of SRRTP-DEC. This instance is a special case of the general single RRT problem. ⊓ ⊔ Easton [31] proofs NP-completeness of a quite similar problem. The problem, namely SRRTP, is to complete a partial single RRT. Easton [31] shows NP-completeness even if each but three periods are scheduled and each team has no more than three unscheduled matches. This problem can be easily reduced to SRRTP-DEC. The idea is to represent a match (i, j, p) which is scheduled in the problem in Easton [31] by forbidding all matches (i′ , j, p), i′ = i, and (i, j ′ , p), j ′ = j, in SRRTP-DEC. Although PTIAP-DEC is not needed for proofing NP-completeness of the single RRT problem the close relation of both problems’ structures is interesting. We make further use of it in section 2.4. Although the single RRT problem is NP-hard feasible solutions can be found in polynomial time. Well known constructions schemes from graph theory have been adapted. An ordered 1-factorization of the complete graph Kn corresponds to a single RRT. See figure 2.1, for example.

2.1 Single Round Robin Tournament i1

i2

i3

i4

11

Fig. 2.1. 1-Factorization of K4

The graph’s vertices represent teams while edge e represents a match between the teams e is incident to. Edges being in the same 1-factor represent matches in the same MD. Edges in figure 2.1 sketched as dotted, dashed, and solid lines form a MD each. Construction schemes for 1-factorizations of complete graphs are known and can be employed in order to generate single RRTs, see Bartsch [5]. Furthermore, heuristics have been developed which randomly construct 1-factorization within seconds, see Dinitz and Stinson [28] and Hilton and Johnson [46] for example. However, these techniques only cover a quite small part of the solution space and, hence, do not suffice to find optimal or even good solutions. Another result from design theory gives an idea about the difficulties arising while tackling the solution space of single RRT problems. Two 1-factorizations f1 and f2 of Kn are said to be isomorphic if there is a mapping φ : V → V such that φ(f1 ) = f2 , see Dinitz et al. [29] and Ihrig [47]. Obviously, φ(f1 ) has the same structure as f1 . The number of classes of non isomorphic 1-factorizations fnc of Kn corresponding to single RRTs having different structures is rarely known: f2c = f4c = c = 396 and no less than f6c = 1, f8c = 6. Gelling and Odeh [43] state f10 c = 20 years later Dinitz et al. [29] report the number for K12 to be f12 c 526, 915, 620. Today fn = is known neither in general nor for n > 12. The number of distinct 1-factorizations fnd is 252,282,619,805,368,320 for n = 12 and is estimated in Dinitz et al. [29] to rise to 1.52 × 1063 for n = 18. Further results can be found in Lindner [56]. So, the number of feasible solutions of the single RRT problem is very large and seems to be an excellent example for combinatorial explosion. For problem sizes larger than n = 12 we do not even know the number of solutions of which we are searching for the best one.

12

2 Basic Problems

2.2 Double Round Robin Tournament Double RRTs are special cases of RRTs and, therefore, have structures according to the requirements presented in section 1.3. Here, r is set to 2 which means that each team meets each other team exactly twice. The matches are to be carried out at different venues. Consequently, the tournament consists of 2 (n − 1) MDs. There are different kinds of double RRTs. We distinguish between mirrored double RRTs, double RRTs based on rounds, and (general) double RRTs below. A mirrored double RRT is a double RRT hosting a match between teams i and j, i, j ∈ T , i = j, in period p, p ∈ P , at i’s home if and only if a match between teams i and j, i, j ∈ T , i = j, at j’s home takes place in period ((t + n − 1) mod (2n − 2)) , that is, the tournament is divided into two single RRTs being complementary to each other; see table 2.2 for an example with n = 6. Table 2.2. Mirrored double RRT for n = 6 period

1.1

1.2

1.3

1.4

1.5

2.1

2.2

2.3

2.4

2.5


5-6 1-4 2-3

3-4 2-5 1-6

4-5 3-1 2-6

5-1 4-2 3-6

2-1 3-5 6-4

6-5 4-1 3-2

4-3 5-2 6-1

5-4 1-3 6-2

1-5 2-4 6-3

Most real world sports leagues are scheduled using the mirrored double RRT scheme. Because of its equivalence to single RRTs we will focus on these in chapter 3 where real world requirements are considered. Similar to the concept in section 2.1 we define the mirrored double RRT problem. Definition 2.5. Given a set T , |T | = n′ , two sets P1 and P2 , |P1 | = |P2 | = n′ − 1, and costs c′i,j,p for each (i, j, p) ∈ T × T × (P1 ∪ P2 ) , i = j, a feasible solution to the mirrored double RRT problem corresponds to a set of n′ (n′ − 1) triples such that (i) for each pair (i, j) ∈ T × T, i < j, exactly one triple of form (i, j, p1 ) or (j, i, p1 ) with p1 ∈ P1 is chosen, (ii) for each pair (i, p1 ) ∈ T × P1 exactly one triple of form (i, j, p1 ) or (j, i, p1 ) with j ∈ T \{i} is chosen, and (iii) for each chosen triple (i, j, p) ∈ T ×T ×P1 , i = j, the triple (j, i, p) ∈ T ×T ×P2 is chosen. The goal of the mirrored double RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Conditions (i) and (ii) ensure a single RRT in P1 and condition (iii) let the matches in P2 be complementary to the ones in P1 . Obviously,

2.2 Double Round Robin Tournament

13

the mirrored double RRT problem can be solved by solving a single RRT problem where cost ci,j,p are defined as follows: ci,j,p = c′i,j,p1(p) + c′j,i,p2(p)

∀i, j ∈ T, i = j, p ∈ P

(2.10)

Note that pr (p) denotes the period of Pr corresponding to index p of P in a single RRT. Theorem 2.2. The mirrored double RRT problem is NP-hard. Proof. We reduce the single RRT problem to the mirrored double RRT problem. The single RRT problem is known to be NP-hard due to theorem 2.1. Given an instance of single RRT problem we construct an instance of mirrored double RRT problem by choosing n′ = n and setting the costs c′i,j,p as follows. ci,j,p ∀i, j ∈ T, i = j, p ∈ P1 ′ ci,j,p = 0 ∀i, j ∈ T, i = j, p ∈ P2 Let td be an optimal solution to the mirrored double RRT problem. Then, the single RRT ts arranged in P1 of td is an optimal solution to the single RRT problem. Suppose there is a single RRT t¯s having lower cost than ts . Then, we can construct a mirrored double RRT t¯d having lower cost than td : set the matches in P1 according to t¯s and let the matches in P2 be complementary to those in P1 . Since t¯s has lower cost than ts and matches in P2 do not affect the tournament’s overall cost t¯d has lower cost than td . ⊓ ⊔ The concept of round-based double RRTs is a generalization of mirrored double RRTs. Here, both rounds might not be complementary. The UEFA champions league is a real world example for round-based double RRTs. Table 2.3 illustrates an example with n = 6 teams. Table 2.3. Round-based double RRT for n = 6 period

1.1

1.2

1.3

1.4

1.5

2.1

2.2

2.3

2.4

2.5


5-6 1-4 2-3

3-4 2-5 1-6

4-5 3-1 2-6

5-1 4-2 3-6

3-2 6-4 1-5

6-1 5-2 4-3

5-4 6-3 2-1

6-5 2-4 1-3

6-2 3-5 4-1

14

2 Basic Problems

We define the corresponding round-based double RRT problem in the following. Definition 2.6. Given a set T , |T | = n′′ , two sets P1 and P2 , |P1 | = |P2 | = n′′ −1, and costs c′′i,j,p for each (i, j, p) ∈ T ×T ×(P1 ∪ P2 ), i = j, a feasible solution to the round-based double RRT problem corresponds to a set of n′′ (n′′ − 1) triples such that (i) for each pair (i, j) ∈ T × T , i < j, exactly one triple of form (i, j, p1 ) or (j, i, p1 ) with p1 ∈ P1 is chosen and (ii) exactly one triple of form (i, j, p) with p ∈ (P1 ∪ P2 ) is chosen, and such that (iii) for each pair (i, p) ∈ T × (P1 ∪ P2 ) exactly one triple of form (i, j, p) or (j, i, p) with j ∈ T \{i} is chosen. The goal of round-based double RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Conditions (i) and (ii) imply that each team plays against each other team exactly once in each round (once at home and once away). Condition (iii) assures that each team plays exactly once in each period. We represent the round-based double RRT problem as an IP model constraints. employing 2n(n − 1)2 variables and 7n(n−1) 2

Model 2.3: RBDRRTP-IP min

c′′i,j,p xi,j,p

(2.11)

i∈T j∈T \{i} p∈(P1 ∪P2 )

s.t.

(xi,j,p1 + xj,i,p1 )

=

1

∀ i, j ∈ T, i < j

(2.12)

=

1

∀ i, j ∈ T, i = j

(2.13)

=

1

∀ i ∈ T, p ∈ (P1 ∪ P2 )

(2.14)

∈

{0, 1} ∀ i, j ∈ T, i = j, p ∈ (P1 ∪ P2 ) (2.15)

p1 ∈P1

xi,j,p

p∈(P1 ∪P2 )

(xi,j,p + xj,i,p )

j∈T \{i}

xi,j,p

Equations (2.12) and (2.14) force the arranged matches to form a single RRT in P1 corresponding to conditions (i) and (iii). Constraint (2.13) ensures that each pair of teams i and j compete twice at different venues representing condition (ii). Hence, taking into account equation (2.14) there is another single RRT formed in P2 .

2.2 Double Round Robin Tournament

15

Theorem 2.3. The round-based double RRT problem is NP-hard. Proof. Reduction of single RRT problem to the round-based double RRT problem with n′′ = n can be done exactly as reduction of single RRT problem to the mirrored double RRT problem. ⊓ ⊔ A (general) double RRT has no additional requirements compared to those specified in section 1.3 with r set to 2, see table 2.4 for example. Table 2.4. Double RRT for n = 6 period

1


2

3

4

5

6

7

8

9

10

1-2 5-3 4-6

5-6 1-4 2-3

3-4 2-5 1-6

6-5 2-4 1-3

4-5 3-1 2-6

5-1 4-2 3-6

3-2 6-4 1-5

6-1 5-2 4-3

6-2 3-5 4-1

We define the corresponding double RRT problem below. Definition 2.7. Given a set T , |T | = n′′′ , a set P , |P | = 2 (n′′′ − 1), and costs c′′′ i,j,p for each (i, j, p) ∈ T × T × P, i = j, a feasible solution to the double RRT problem corresponds to a set of n′′′ (n′′′ − 1) triples such that (i) for each pair (i, j) ∈ T × T, i = j, exactly one triple of the form (i, j, p) with p ∈ P is chosen and such that (ii) for each pair (i, p) ∈ T × P exactly one triple of form (i, j, p) or (j, i, p) with j ∈ T \{i} is chosen. The double RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. We represent the double RRT problem as an IP model using 2n(n − 1)2 variables and 2n(n − 1) constraints, see (2.16) to (2.19). (2.17) and (2.18) directly correspond to (i) and (ii), respectively. (2.16) states the goal to minimize arranged matches’ costs. Theorem 2.4. The double RRT problem is NP-hard. Proof. We reduce the single RRT problem to the double RRT problem. The single RRT problem is known to be NP-hard due to theorem 2.1. Given an instance of the single RRT problem we construct an instance of the double RRT problem with n′′′ = n as follows. Let f be an arbitrarily ordered 1-factorization of Kn , built using the method presented in Schreuder [76] for example. Remember that there is always an 1-factorization of Kn if n is even. We define costs c′′′ i,j,p of the double RRT problem as follows:

16

2 Basic Problems

Model 2.4: DRRTP-IP min

c′′′ i,j,p xi,j,p

(2.16)


s.t.

xi,j,p

=

1

∀ i, j ∈ T, i = j

(2.17)

=

1

∀ i ∈ T, p ∈ P

(2.18)

∈

{0, 1} ∀ i, j ∈ T, i = j, p ∈ P

(2.19)

p∈P

(xi,j,p + xj,i,p )

j∈T \{i}

xi,j,p

 ci,j,p ∀ i, j ∈ T, i = j, p ∈ {1, . . . , n − 1}      0 ∀ (i, j, n − 1 + p) with (i, j), i < j,    being in 1-factor p of f c′′′ i,j,p =  0 ∀ (j, i, n − 1 + p) with (i, j), i < j,     being in 1-factor p of f     M otherwise with M = i∈T j∈T \{i} p∈P ci,j,p . Let td be an optimal solution to the double RRT problem. Then, the single RRT ts arranged in periods {1, . . . , n − 1} of td is an optimal solution to the single RRT problem. Suppose there is a single RRT t¯s having lower cost than ts . Then, we can construct a double RRT t¯d having lower cost than td : set the matches in periods {1, . . . , n − 1} according to t¯s . Additionally, let the matches in periods {n, . . . , 2n − 2} correspond to f . Then, matches in periods {n, . . . , 2n − 2} have overall cost of zero which, obviously, is the minimum possible value. Clearly, t¯d is a (general) double RRT and, furthermore, t¯d has lower cost than td has since t¯s has lower cost than ts . ⊓ ⊔

2.3 r Round Robin Tournament An obvious generalization of the RRTs presented so far is to let r ∈ N, in particular r > 2. We take into account the same interrelations between rounds as outlined in section 2.2. In the following we describe the resulting RRTs and give corresponding cost minimization problems

2.3 r Round Robin Tournament

17

all of which are NP-hard. We renounce to give proofs of complexity and IP models, respectively, since they are straightforward from those in section 2.2. A mirrored r-RRT is a r-RRT hosting a match between teams i ∈ T and j ∈ T , i = j, at i’s home in period p ∈ {1, . . . , (r − 1)(n − 1)} if and only if a match between teams i and j at j’s home takes place in period p + n − 1. Hence, the tournament is divided into r single RRTs where single RRT r ′ , r ′ ∈ {1, . . . , r − 1}, is complementary to single RRT r ′ + 1. We define the corresponding cost minimization problem hereafter. Definition 2.8. Given a set T with |T | = n′ , sets Pr′ with |Pr′ | = (n′ − 1) for each r ′ ∈ {1, . . . , r}, and costs c′i,j,p for each (i, j, p) ∈ T × T × ∪rr′ =1 Pr′ , i = j, a feasible solution to the mirrored r-RRT ′ ′ problem corresponds to a set of r n (n2 −1) triples such that (i) for each pair (i, j) ∈ T × T, i < j, exactly one triple of form (i, j, p1 ) or (j, i, p1 ) with p1 ∈ P1 is chosen, (ii) for each pair (i, p1 ) ∈ T × P1 exactly one triple of form (i, j, p1 ) or (j, i, p1 ) with j ∈ T \{i} is chosen, and (iii) for each chosen triple (i, j, p) ∈ T × T × Pr′ , i = j, r ′ ∈ {1, . . . , r − 1}, the triple (j, i, p) ∈ T × T × Pr′ +1 is chosen. The goal of the mirrored r-RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Conditions (i) and (ii) ensure that a single RRT takes place in the first round. Condition (iii) induces each round to be complementary to the previous and the following one. Obviously, mirrored r-RRT problems can be solved by solving single RRT problems. To this end, we set cost of a single RRT problem according to a generalization of equation (2.10):

ci,j,p =

⌈ 2r ⌉

r ′ =1

c′i,j,p ′ (p) 2r −1

+

⌊ 2r ⌋

r ′ =1

c′j,i,p

2r ′ (p)

∀i, j ∈ T, i = j, p ∈ P (2.20)

A round-based r-RRT is a r-RRT according to characteristics introduced in section 1.3. Its periods can be partitioned into r rounds where each round is a single RRT. We define the corresponding cost minimization problem below. Definition 2.9. Given a set T with |T | = n′′ , sets Pr′ with |Pr′ | = (n′ − 1) for each r ′ ∈ {1, . . . , r}, and costs c′′i,j,p for each (i, j, p) ∈ T × T × ∪rr′ =1 Pr′ , i = j, a feasible solution to the round-based r-RRT

18

2 Basic Problems ′′

′′

problem corresponds to a set of r n (n2 −1) triples such that (i) for each r ′ ∈ {1, . . . , r} the chosen triples (i, j, p) with p ∈ Pr′ form a single

RRT and such that (ii) for each pair (i, j) ∈ T × T, i = j, at least 2r triples (i, j, p) with p ∈ ∪rr′ =1 Pr′ are chosen. The goal of the round-based r-RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Condition (i) is stated straightforwardly. Condition (ii) limits the difference of number of matches between teams i ∈ T and j ∈ T , i = j, at i’s home and number of matches between teams i and j at j’s home to be no more than 1. A (general) r-RRT is fully specified by the characteristics given in section 1.3. In the following, the cost minimization problem is given. Definition 2.10. Given a set T with |T | = n′′′ , a set P with |P | = r(n′′′ − 1), and costs c′′′ i,j,p for each (i, j, p) ∈ T × T × P, i = j, a feasible ′′′

′′′

solution to the r-RRT problem corresponds to a set of r n (n2 −1) triples such that (i) for each pair (i, j) ∈ T ×T, i = j, at least 2r triples (i, j, p) with p ∈ P are chosen, such that (ii) for each pair (i, j) ∈ T × T, i < j, exactly r triples (i, j, p) or (j, i, p) with p ∈ P are chosen and such that (iii) for each pair (i, p) ∈ T × P exactly one triple of form (i, j, p) or (j, i, p) with j ∈ T \{i} is chosen. The goal of r-RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Condition (i) limits the difference of number of matches between teams i ∈ T and j ∈ T , i = j, at i’s home and number of matches between teams i and j at j’s home to be no more than 1. Condition (ii) implies that each pair of teams meets exactly r times. Condition (iii) assures that each team has exactly one opponent per period. Note that we can drop (ii) if r is even, since then (ii) is implied by (i).

2.4 Decomposition Schemes Due to the complexity of RRT problems outlined in sections 2.1 to 2.3 two decomposition schemes are used frequently. Both separate the decisions about the period a match takes place in and about the venue it is carried out at. • First–Schedule–Then–Break: First, each pair of teams is fixed to compete in a specific period. Based on this timetable each match’s venue is determined, see Trick [81] for example.

2.4 Decomposition Schemes

19

• First–Break–Then–Schedule: First, the matches’ venues are decided. Afterwards, the matches are assigned to periods, see Nemhauser and Trick [68] for an example. Optimization problems considered below refer to single RRTs. However, adaption to other basic problems is straightforward. 2.4.1 First–Schedule–Then–Break An opponent schedule, as defined in Post and Woeginger [69], is a timetable which determines for each pair (i, p), i ∈ T , p ∈ P , the opponent of team i in period p. See table 2.5 for an example corresponding to the single RRT in table 2.1. Team i’s opponent in period p is specified in the row corresponding to i in column p. Table 2.5. Opponent Schedule for n = 6 team 1 1 2 3 4 5 6

2 1 5 6 3 4

2

3

4

5

4 3 2 1 6 5

6 5 4 3 2 1

3 6 1 5 4 2

5 4 6 2 1 3

In the following, we formally define the problem to find the minimum cost opponent schedule. Definition 2.11. Given a set T , |T | = n′ , a set P , |P | = n′ − 1, and costs c′i,j,p for each (i, j, p) ∈ T × T × P , i < j, a feasible solution to ′

′

the opponent-schedule problem corresponds to a set of n (n2−1) triples such that (i) for each subset of teams (i, j) ∈ T × T , i < j, exactly one triple (i, j, p) with p ∈ P is chosen and such that (ii) for each pair (i, p) ∈ T ×P exactly one triple of form (i, j, p), j ∈ T , i < j, or (j, i, p), j ∈ T , j < i, is chosen. The goal of the opponent-schedule problem is to find a feasible solution having the minimum sum of chosen triples’ cost. Condition (i) ensures that each team meets each other team while condition (ii) forces each team to compete exactly once per period.

20

2 Basic Problems

These requests as well as the cost minimization goal can be repre2 sented as an IP model employing n(n−1) binary variables and 3n(n−1) 2 2 constraints, see (2.21) to (2.24).

Model 2.5: Opponent-IP min

c′i,j,p x′i,j,p

(2.21)

i∈T i<j p∈P

s.t.

x′i,j,p

p∈P

x′i,j,p +

i<j

x′i,j,p

x′j,i,p

=

1

∀ i, j ∈ T, i < j

(2.22)

=

1

∀ i ∈ T, p ∈ P

(2.23)

∈

{0, 1} ∀ i, j ∈ T, i < j, p ∈ P

(2.24)

j

Binary variable x′i,j,p, i, j ∈ T , i < j, p ∈ P , is equal to 1 if and only if teams i and j compete in p. The objective to minimize the cost corresponding to the opponent schedule is given in (2.21). Restriction (2.22) assures that each pair of teams meet and constraint (2.23) forces each team to play exactly once per period. Theorem 2.5. The opponent-schedule problem is NP-hard. Proof. We reduce the single RRT problem to the opponent-schedule problem by setting costs c′i,j,p as follows: c′i,j,p = min{ci,j,p , cj,i,p } ∀ i, j ∈ T, i < j, p ∈ P Given an optimal solution to to the opponent-schedule problem we can construct an optimal solution ts to the corresponding single RRT problem as follows: • For each chosen (i, j, p), i < j, of the opponent-schedule problem choose (i, j, p) of the single RRT problem if ci,j,p ≤ cj,i,p. • For each chosen (i, j, p), i < j, of the opponent-schedule problem choose (j, i, p) of the single RRT problem if ci,j,p > cj,i,p. Obviously, ts is a single RRT. Furthermore, ts is optimal to the single RRT problem. Suppose there is a solution t¯s to the single RRT problem having lower cost than ts . Then, there is a solution t¯o := {(i, j, p) | i < j, (i, j, p) ∈ t¯s ∨ (j, i, p) ∈ t¯s } to the opponentschedule problem having lower cost than to . ⊓ ⊔


21

In the course of the first-schedule-then-break scheme the venue of each match is determined next. The venue can be specified by assigning a home-away-pattern (HAP) to each team. Definition 2.12. A HAP is a string of length n − 1 containing 0 at slot p if the specific team plays at home in period p, 1 otherwise. Definition 2.13. A HAP set is a collection of HAPs where exactly one HAP is assigned to each team. Table 2.6 illustrates the HAP set corresponding to the single RRT shown in table 2.1. Table 2.6. HAP set for n = 6 team 1 1 2 3 4 5 6

0 1 1 0 0 1

2

3

4

5

0 0 1 1 0 1

0 0 0 1 1 1

1 0 0 0 1 1

1 1 0 0 0 1

Let h be a HAP set and let hi,p be the entry corresponding to team i ∈ T and period p ∈ P . Then, h is called feasible for an opponent schedule o if for each team i competing team j in period p according to o entries hi,p and hj,p are not identical. Note that the HAP set presented in table 2.6 is feasible to the opponent schedule given in table 2.5. Probably, given an opponent schedule the most popular goal when determining the matches’ venues is to find the HAP set which is feasible to the opponent schedule and has the minimum number of breaks. A break occurs if a team plays twice at home or twice away in two consecutive periods. The resulting break-minimization problem is focus of many research activities. It is known from de Werra [19] that the number of breaks can not be less than n − 2. Elf et al. [36] conjecture the break-minimization problem to be NP-hard in general. Miyashiro and Matsui [63] and Miyashiro and Matsui [62] proof the problem of either finding a HAP set with the minimum number of n − 2 breaks if one exists or deciding there is no such HAP set to be solvable in polynomial time. The same is true for equitable HAP sets assigning exactly one break to each team.

22

2 Basic Problems

Post and Woeginger [69] consider partial opponent schedules containing less than n − 1 periods. They state that the break minimization problem for partial opponent schedules is NP-hard if the number of periods is greater than 2. 2.4.2 First–Break–Then–Schedule First, for each team i ∈ T and each period p ∈ P it is decided whether i plays at home or away in p. Reasonably, a HAP set can be constructed considering the preferences of teams to play at home or away, for example. Furthermore, stadium availability and fixed matches can be incorporated. The resulting HAP set restricts the following scheduling process: no team playing at home or away in period p according to the HAP set can play away or at home, respectively, in the final SLS. Note that, consequently, two teams i ∈ T and j ∈ T can not compete in period p if hi,p = hj,p . Obviously, there is no guarantee that a random HAP set h ∈ {0, 1}n×(n−1) allows one single RRT to be arranged. In the remainder, a HAP set h is called feasible if and only if at least one RRT can be scheduled based on h. We define the corresponding decision problem. Definition 2.14. We define the HAP set feasibility problem as follows: Input: A HAP set h ∈ {0, 1}n×(n−1) . Question: Is h feasible? There are two obvious necessary conditions for a HAP set h to be feasible: (i) HAPs of two teams can not be identical. If two HAPs are identical the corresponding teams can not play against each other in any period. (ii)Each column of h has to contain exactly n2 zeros. If this does not hold for a specific period p not each team can play in p. Unfortunately, (i) and (ii) are not sufficient for h to be feasible. Miyashiro et al. [64] provide additional necessary condition (2.25) for HAP sets to be feasible. Let c0 (T ′ , p) and c1 (T ′ , p) be the number of zeros and ones, respectively, corresponding to a subset T ′ ⊆ T of teams and column p.

p∈P

′

|T | ′ ′ min c0 (T , p), c1 (T , p) − ≥ 0 ∀T′ ⊆ T 2

(2.25)


23

Condition (2.25) states that each subset T ′ of teams must be allowed ′ to play |T2 | matches against each other. The number of matches between teams of T ′ in each period is restricted to the minimum of numbers of teams in T ′ playing at home and away, respectively. Note that condition (2.25) is a generalization of (ii). As shown in Miyashiro et al. [64] for HAP sets having the minimum number of breaks condition (2.25) can be checked in polynomial time and, moreover, it is conjectured to be sufficient. However, sufficiency is not proven and, therefore, complexity of the HAP set feasibility problem is open so far for arbitrary HAP sets as well as for HAP sets having minimum number of breaks. We propose a condition for HAP sets to be feasible which can be checked in polynomial time. First, we formulate IP model HAP-setfeasibility in order to check feasibility of a given HAP set h as proposed in Briskorn [8]. Again, binary variable x′i,j,p, i, j ∈ T , j < i, p ∈ P , is equal to 1 if and only if teams i and j compete in period p.

Model 2.6: HAP-set-feasibility max zh = x′i,j,p

(2.26)

i∈T j∈T,ji

≤ |hi,p − hj,p | ∀ i, j ∈ T, j < i, p ∈ P (2.29) ∈ {0, 1}

∀ i, j ∈ T, j < i, p ∈ P (2.30)

Objective function (2.26) represents the goal to maximize the number of matches while constraints (2.27) and (2.28) assure a single RRT. More precisely, constraint (2.27) forces each pair of teams to meet at most once while constraint (2.28) restricts the number of matches per team and period to be less than or equal to one. The entry of HAP set h corresponding to team i and period p is denoted by hi,p . Then, (2.29)

24

2 Basic Problems

takes care of the HAP set h such that no pair of teams can meet in a period where both of them play at home or away, respectively. Clearly, h is feasible if and only if zh = n(n−1) . Let z¯h be the max2 imum objective value to the linear programming (LP) relaxation of is a necHAP-set-feasibility. Then, z¯h ≥ zh and, hence, z¯h = n(n−1) 2 essary condition for h to be feasible. Furthermore, we can decide in polynomial time whether z¯h = n(n−1) or not, see Garey and Johnson 2 [42]. Theorem 2.6. Condition z¯h =

n(n−1) 2

is strictly stronger than (2.25).

Proof. First, we show that each HAP set h being infeasible according to condition (2.25) is infeasible according to condition z¯h = n(n−1) , as well. Suppose z¯h = n(n−1) . We count the matches be2 2 ′ T ′ . Obviously, T′ = tween teams in T ⊆ T in period p as z¯h,p ¯h,p p∈P z |T ′ |(|T ′ |−1) . 2

′

T and, thus, Moreover, min {c0 (T ′ , p), c1 (T ′ , p)} ≥ z¯h,p ′ |T | ′ ′ holds. p∈P min {c0 (T , p), c1 (T , p)} ≥ 2 Second, we consider a HAP set h provided by Kashiwabara [49] with n = 14 shown in table 2.7. HAP set h fulfills (2.25) but has z¯h = 90 < 91 = n(n−1) . 2

Table 2.7. Infeasible HAP set for n = 14 team 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 0 0 0 1 1 1 1 1 0 1 1

2

3

4

5

6

7

8

9 10 11 12 13

0 0 0 0 0 0 1 1 1 1 1 0 1 1

0 0 0 0 0 0 1 1 1 1 1 1 0 1

0 0 0 0 0 0 1 1 1 1 1 1 1 0

1 1 1 1 1 0 0 0 0 0 0 0 1 1

1 1 1 1 1 0 0 0 0 0 0 1 0 1

1 1 1 1 1 0 0 0 0 0 0 1 1 0

1 1 1 1 1 0 0 0 0 0 0 1 1 0

0 0 1 1 1 1 0 0 1 1 1 0 0 0

1 0 0 1 1 1 1 0 0 1 1 0 0 0

1 1 0 0 1 1 1 1 0 0 1 0 0 0

1 1 1 0 0 1 1 1 1 0 0 0 0 0

0 1 1 1 0 0 0 1 1 1 0 0 1 0


Consequently, h is infeasible which is detected by z¯h < not by (2.25).

n(n−1) 2

25

but ⊓ ⊔

Moreover, we conjecture z¯h = zh for each h ∈ {0, 1}n×(n−1) . Unfortunately, we can not provide a formal proof. In order to support the conjecture we have carried out computational tests for two different classes of HAP sets solving HAP-set-feasibility and the corresponding LP relaxation. Given n these classes of HAP sets are generated as follows: I Each entry is randomly chosen from {0, 1}. II For each slot p we randomly choose n2 teams having 0 in p. The remaining teams have 1 in p. Obviously, for HAP sets according to class II the probability to be feasible is much higher since identical amounts of zeros and ones in each slot are necessary for a HAP set to be feasible. We carried out test runs for 10.000 instances for each class and each n ∈ {6, . . . , 30}. Summarizing the results of the experiments our conjecture is true for each of the 260.000 considered instances. If z¯h = zh holds for each h ∈ {0, 1}n×(n−1) then z¯h = n(n−1) is not 2 only a necessary but also a sufficient condition for h to be feasible. Then, the HAP set feasibility problem is solvable in polynomial time via solving the LP relaxation of HAP-set-feasibility to optimality. In the course of the first-break-then-schedule scheme an opponent schedule according to the given HAP set is determined next. We specify this in the following cost minimization problem. Definition 2.15. Given a feasible HAP set h, a set T , |T | = n, a set P , |P | = (n − 1), and costs ci,j,p for each (i, j, p) ∈ T × T × P, i = j, hi,p = 0, hj,p = 1, a feasible solution to the HAP set-based RRT problem corresponds to a set of |T |(|T2 |−1) triples forming a single RRT. The goal of HAP set-based RRT problem is to find a feasible solution having the minimum sum of chosen triples’ cost. We add constraint (2.31) to SRRTP-IP problem in order to represent the HAP set-based RRT problem. Theorem 2.7. The HAP set-based RRT problem is NP-hard. Theorem 2.7 is proven by reduction from PTIAP to the HAP-setbased RRT problem. The proof is based on the same idea as the reduction of PTIAP-DEC to SRRTP-DEC is.

26

2 Basic Problems

Model 2.7: HAP–set–based SRRTP-IP xi,j,p

=

0

∀ i, j ∈ T, i = j, p ∈ P, hi,p = 1 ∨ hj,p = 0

(2.31)

Proof. Given a PTIAP of size m, m even, we construct an instance of the HAP-set-based RRT problem with n = 2m as follows. First, we define a HAP set h of HAP set-based RRT problem. We follow the idea of the binary 1-factorization as proposed in de Werra [19] and let 0 if i ∈ {1, . . . , m} , p ∈ {1, . . . , m} , hi,p = 1 if i ∈ {m + 1, . . . , 2m} , p ∈ {1, . . . , m} . Additionally, we set hi,p , p ∈ {m + 1, . . . , 2m − 1}, according to two arbitrary single RRTs for subsets of teams T1 := {1, . . . , m} and T2 := {m + 1, . . . , 2m} in periods p ∈ {m + 1, . . . , 2m − 1}. Such single RRTs always exist since m is even. Now, a HAP set for 2m teams is fully specified. Construction of costs ci,j,p is done as follows:  di,j−m,p if i, p ∈ {1, . . . , m} , j ∈ {m + 1, . . . , 2m} ,      M if i ∈ {1, . . . , m} , j ∈ {m + 1, . . . , 2m} ,     p ∈ {m + 1, . . . , 2m − 1} , hi,p = 0, hj,p = 1,      if i, j ∈ {1, . . . , m} , i = j, p ∈ {m + 1, . . . , 2m − 1} , 0 hi,p = 0, hj,p = 1, ci,j,p =     0 if i, j ∈ {m + 1, . . . , 2m} , i = j,     p ∈ {m + 1, . . . , 2m − 1} , hi,p = 0, hj,p = 1,      if i ∈ {m + 1, . . . , 2m} , j ∈ {1, . . . , m} ,  M p ∈ {m + 1, . . . , 2m − 1} , hi,p = 0, hj,p = 1, where M = a∈A b∈B c∈C da,b,c . Each optimal solution to the single RRT does not contain a match having cost M due to costs’ construction. Hence, given an optimal solution ts to the single RRT problem we can create a solution ttia to the PTIAP by choosing triple (i, j − m, p) of PTIAP for each chosen triple (i, j, p) of the single RRT problem. Obviously, ttia is feasible and, moreover, optimal. Suppose there is a solution t¯tia having lower cost than ttia . Then, we can construct a single RRT t¯s having lower cost than


27

ts : we replace triples corresponding to ttia by triples corresponding to t¯tia in ts . ⊓ ⊔

3 Real World Problems

This chapter focuses on IP models representing scheduling problems in real world RRTs as considered in Briskorn and Drexl [9]. We consider several requirements having practical relevance and represent them by means of IP constraints. Additionally, we provide a computational study considering the single RRT problem introduced in chapter 2 and real world requirements presented in the following. We choose SRRTP-IP presented in section 2.1 as structural core for everything what follows. For the sake of convenience we repeat the formulation.

Model SRRTP-IP min

ci,j,p xi,j,p

(2.6)


s.t.

(xi,j,p + xj,i,p )

=

1

∀ i, j ∈ T, i < j

(2.7)

=

1

∀ i ∈ T, p ∈ P

(2.8)

∈

{0, 1} ∀ i, j ∈ T, i = j, p ∈ P

(2.9)

p∈P

(xi,j,p + xj,i,p )

j∈T \{i}

xi,j,p

Choosing single RRTs is justified by the fact that many professional soccer, ice hockey, and handball leagues have schedules according to mirrored double RRTs. We can tackle corresponding problems by solving single RRT as outlined in section 2.2. Consequently, we project re-

30


quirements resulting from the second round of a mirrored double RRT to the unique round of single RRT. Furthermore, there are many mega events, e.g. soccer world cup, having several single RRTs followed by a play-off phase. Adaption to other RRTs presented in sections 2.2 and 2.3 is straightforward. In section 3.1 constraints given by outer league parties are proposed. These constraints cannot be influenced by the league’s organizers or participants, respectively. Constraints in section 3.2 assure some kind of fairness in SLSs and can be applied, modified, or dropped by the organizers’ own decision.

3.1 Externally Given Constraints 3.1.1 Forbidden Matches There are several reasons to leave specific matches (i, j, p) out of consideration. We introduce binary parameter πi,j,p indicating whether match (i, j, p) is possible (πi,j,p = 1) or not (πi,j,p = 0). As outlined in section 1.3 matches are carried out in one of both teams’ stadiums. Although a particular stadium is associated to each team the stadium might not be owned by the club. Often the corresponding city or some public agency is owner or co-owner of the venue. Hence, stadiums generally are used for hosting other events such as pop concerts, too. When scheduling a sports league we can not take stadium availability for granted. Another reason might be the stadium being closed due to construction works. Consequently, in periods where the stadium of team i is not available we have to arrange a match at the opponent’s venue for i. If i’s stadium is not available in period p we set πi,j,p = 0 for all j ∈ T \ {i}. The other way round, i has to play at home in period p if its stadium is not available in the corresponding period of the second round in a mirrored double RRT. Consequently, if i’s stadium is not available in period p of the second round we set πj,i,p = 0 for all j ∈ T \ {i}. Furthermore, we fix i’s stadium as the venue and p as the period where a match between teams i and j is carried out by setting πi,j ′ ,p = 0 for each j ′ ∈ T \ {i, j}, πj ′ ,i,p = 0 for each j ′ ∈ T \ {i}, πi,j,p′ = 0 for each p′ ∈ P \ {p}, and πj,i,p′ = 0 for each p′ ∈ P . Analogously, we can force pairs of teams supposed to be among the best ones to compete in one of the last periods in order to create a thrilling finale phase of tournament. We employ πi,j,p in n(n − 1)2 constraints. Trivially, constraint (3.1) forces xi,j,p to be equal to 0 if match (i, j, p) is not possible.

3.1 Externally Given Constraints

31

Model 3.1: Forbidden Matches (SRRTP-IP) xi,j,p

≤

πi,j,p

∀ i, j ∈ T, i = j, p ∈ P

(3.1)

3.1.2 Regions’ Capacity In real world leagues there are regions in which more than one team is located. Consider, for instance, European professional soccer leagues where some capitals host more than one team. A prominent example is London where no less than 13 professional teams are located. Five of them played in the Premier League, England’s first soccer league, in season 2005/2006. Even if each of these teams has an own stadium the infrastructure of the region might be overloaded if too many teams play at home in a specific period. For example, traffic jams and overcrowded public transportation systems resulting from fans heading to the stadiums at the same time must be avoided. Furthermore, the capacity of security staff and of firemen needed in case of emergency is limited. In order to model the limitation of the number of matches carried out in a specific region in each period we introduce the set R of regions with each R′ ∈ R being a subset of T . In most real world examples |R′ ∩ R′′ | = 0 holds for all R′ , R′′ ∈ R, R′ = R′′ . However, there is a formal need neither for R to exclusively contain pairwise disjoint subsets nor for R to fully cover T . We associate a maximum number of matches CR′ per period with each R′ ∈ R. Parameter CR′ has to be customized for each region taking into account all individual aspects such as |R′ |, region’s expansion, road network, and railway system.

Model 3.2: Regions (SRRTP-IP) xi,j,p ≤ CR′

∀ R′ ∈ R, p ∈ P

(3.2)

xj,i,p ≤ CR′

∀ R′ ∈ R, p ∈ P

(3.3)

i∈R′

j∈T \{i}

i∈R′ j∈T \{i}

When scheduling a single RRT only |R|(n − 1) constraints according to (3.2) are employed. The number of matches in each region R′ is forced to be no more than CR′ in each period. In case of a mirrored double RRT we add constraint (3.3) in order to limit the number of matches

32


in region R′ in period p in the second round by limiting the number of teams of region R′ playing away in period p of the first round. ′ Note that, obviously, CR′ can not be set to a value less than |R2 | for a mirrored double RRT since the sum of left hand sides of restrictions (3.2) and (3.3) sum up to |R′ |. 3.1.3 Highly Attended Matches Broadcasting stations grow more and more important for leagues and teams. This could be clearly observed when Leo Kirch, owner of german broadcasting station “Premiere”, suffered insolvency and as a result many german soccer clubs had a high budget deficit. A match’s attendance depends among others on the two teams competing. Based on, e.g., the score in the previous competition we can identify matches having high attractiveness for spectators and, hence, for broadcasting stations, too. Broadcasting stations are interested in presenting as many attractive matches as possible. If two matches are scheduled at the same time it is not possible to broadcast both of them. Therefore, the number of attractive matches carried out in each period shall be kept below a certain threshold leading to a balanced distribution of attractive matches over all periods. Obviously, planning can be done more accurate if it is based not only on periods {1, . . . , n − 1} but also on day of week and time of day. However, the concept presented here avoids accumulation of to many attractive matches in a single MD. We introduce binary parameter ai,j , i, j ∈ T, i = j, indicating whether a match of team i at home against team j is attractive (ai,j = 1) or not (ai,j = 0). Obviously, in most cases ai,j = aj,i holds. However, ai,j > aj,i might make sense if i is a team having medium strength and j is a top level team, for example. If j plays at home it probably has an easy win but if the match is carried out at i’s venue it might be a close and exciting match. Furthermore, we employ the parameter amax as an upper bound for the number of attractive matches scheduled per period. The limitation is represented by n − 1 constraints (3.4).

Model 3.3: Highly Attended Matches (SRRTP-IP) ai,j xi,j,p ≤ amax ∀ p ∈ P i∈T j∈T \{i}

(3.4)

3.2 Fairness Constraints

33

3.2 Fairness Constraints 3.2.1 Breaks Minimum Number of Breaks Team i is said to have a break in period p if i plays twice at home or twice away, respectively, in consecutive periods p − 1 and p. We distinguish between home-breaks and away-breaks according to the venue of both matches. The most prominent goal considering breaks is minimizing the overall number. In de Werra [19] the number of breaks is shown to be no less than n − 2. Let P ≥2 := P \ {1}. We represent the requirement of the minimum number of breaks by 2n (n − 2) + 1 constraints employing n (n − 2) additional binary variables bri,p , i ∈ T , p ∈ P ≥2 .

Model 3.4: Minimum Number of Breaks (SRRTP-IP) (xi,j,p−1 + xi,j,p ) − bri,p ≤ 1 ∀ i ∈ T, p ∈ P ≥2

(3.5)

∀ i ∈ T, p ∈ P ≥2

(3.6)

j∈T \{i}

j∈T \{i}

(xj,i,p−1 + xj,i,p ) − bri,p

≤

1

bri,p

≤

n−2

(3.7)

bri,p

∈

{0, 1} ∀ i ∈ T, p ∈ P ≥2

(3.8)

i∈T p∈P

Constraints (3.5) and (3.6) force bri,p to be equal to 1 if team i has a home-break or an away-break in period p, respectively. Since we assure the minimum number of breaks by restriction (3.7) bri,p is equal to 1 if and only if i has a break in period p. Additionally, we consider a common requirement in real world leagues: a break must not occur in the second period. The reason for this is that breaks are considered as some kind of disturbance in the season’s regularity. Disturbance shall be avoided at the season’s beginning to guarantee a fair start. Trivially, breaks in the second period can be eliminated by forcing bri,2 = 0 for each i ∈ T .

34


One Break per Team Although minimizing the number of breaks is most popular it is reasonable to think about arranging exactly one break for each team for fairness reasons as proposed in de Werra [19]. This requirement can be represented employing 2n (n − 2) binary variables hbri,p and abri,p , i ∈ T , p ∈ P ≥2 in n (n − 1) constraints (3.9) and (3.10). Variables hbri,p and abri,p are equal to one if and only if i has a home-break and an away-break, respectively.

Model 3.5: One Break per Team (SRRTP-IP) (xj,i,p−1 + xj,i,p ) + hbri,p − abri,p =1 ∀ i ∈ T, p ∈ P ≥2 j∈T \{i}

(hbri,p + abri,p ) =1

∀i ∈ T

(3.9) (3.10)

p∈P ≥2

hbri,p ∈ {0, 1} ∀ i ∈ T, p ∈ P ≥2 (3.11) abri,p ∈ {0, 1} ∀ i ∈ T, p ∈ P ≥2 (3.12)

In order to illustrate constraint (3.9) we consider three cases. If team i has a home-break in period p the sum of match variable values equals 0 and, therefore, hbri,p and abri,p are forced to 1 and 0, respectively. If i has an away-break in period p match variable values sum up to 2 letting hbri,p = 0 and abri,p = 1. If i has no break in period p, finally, the sum of match variable values equals 1. Hence, hbri,p = abri,p = 0 follows taking constraint (3.10) into account. Trivially, constraint (3.10) forces exactly one break for each team i ∈ T . 3.2.2 Opponents’ Strengths Teams have different strengths, indicated for instance through the final score obtained in the previous season. Hence, two matches of team i ∈ T might be distinctly exhausting for i depending on the particular opponents. Assume that the set of teams T is partitioned by a set of strength groups S. Each team is contained group in exactly one strength ′ ′ s ∈ S. Thus s ⊂ T , |s ∩ s | = 0, and s∈S s = T with s, s ∈ S, s = s′ , hold. For the sake of convenience we assume that |T | mod |S| = 0 and that |s| = |s′ | for all s, s′ ∈ S. Furthermore,each strength group s ∈ S n n . + 1, . . . , s |S| is specified by s = (s − 1) |S|


35

In order to avoid a burden of competition considered to be too high (and hence unfair) for some teams we claim that the matches of each team against teams of a specific strength group shall be distributed as even as possible over the tournament. Hereafter we present four different ways to enforce different degrees of fairness considering the idea of strength groups. An interesting question from a combinatorial point of view arises for each of these structures: Given a number of strength groups |S| and a number of teams n is it possible to arrange a single RRT obeying one of the fairness constraints outlined in the following? This question is addressed in detail in chapter 4. Changing Strength Groups Since we aim at distributing matches of team i ∈ T against teams of the same strength group equally over periods 1, . . . , n−1 the worst case corresponds to two matches of team i in consecutive periods against teams belonging to the same strength group. See table 3.1 for an example considering team i = 1 in a league with 8 teams and 4 strength groups. The second and third row shows matches team i = 1 contributes to and the corresponding opponent’s strength group, respectively. Apparently, there is no change in the opponent’s strength group between periods 3 and 4. We employ n(n − 2) binary variables esci,p , i ∈ T , p ∈ P ≥2 , Table 3.1. Opponents’ Strength Groups Not Changing period

1

2

3

4

5

6

7

match s

1-2 1

7-1 4

1-5 3

6-1 3

1-3 2

8-1 4

1-4 2

to indicate such a case, that is, esci,p = 1 if team i ∈ T plays against teams of the same strength group in periods p − 1 and p. Constraint (3.13) forces binary variable esci,p to be equal to 1 if team i has two opponents of the same strength group in periods p − 1 and p. Inequality (3.14) assures that the number of violations of the changing strength group requirement is limited to a given parameter escmax representing the maximum number of allowed occurrences. Table 3.2 represents a feasible timetable for i = 1 if we consider escmax = 0. The solution illustrated in table 3.1 is feasible if and only if escmax > 0 holds.

36


Model 3.6: Changing Strength Groups (SRRTP-IP) (xi,j,p−1 + xj,i,p−1 +

(3.13)

j∈s,j =i

xi,j,p + xj,i,p ) − esci,p esci,p

≤

1

∀ i ∈ T, s ∈ S, p ∈ P ≥2

≤

escmax

∀i ∈ T

(3.14)

∈

{0, 1}

∀ i ∈ T, p ∈ P ≥2

(3.15)

p∈P ≥2

esci,p

Table 3.2. Opponents’ Strength Groups Changing period

1

2

3

4

5

6

7

match s

3-1 2

1-2 1

4-1 2

5-1 3

1-7 4

6-1 3

1-8 4

Balanced Strength Groups Obviously, the timetable given in table 3.2 does not provide a perfect distribution of opponents’ strength groups over the tournament. Only teams of groups s = 1 and s = 2 are opponents in periods 1 to 3, opponents of groups s = 3 and s = 4 are restricted to periods 4 to 7. In order to get a more balanced distribution we present a formulation claiming that the number of occurrences of each strength group as opponent of i is restricted to no more than 1 in a time window containing |S| consecutive periods. Clearly, the requirement is violated if there is any time window [p, p + |S| − 1], p ∈ {1, . . . , n − |S|}, containing two opponents of the same strength group s ∈ S for team i ∈ T . Let j ∈ T and j ′ ∈ T , j ′ = j, be teams of the same strength group competing i in periods p′ and p′′ , respectively. Note that the violation is the more severe the smaller |p′ − p′′ | is. For instance consider table 3.3. Table 3.3. Opponents’ Strength Groups Not Balanced period

1

2

3

4

5

6

7

match s

7-1 4

2-1 1

3-1 2

1-8 4

4-1 2

1-5 3

1-6 3


37

Team i’s opponents belong to strength group s = 2 in periods 3 and 5. Note that this affects two time windows of length |S|: [2, 5] and [3, 6]. Furthermore, s = 4 appears in periods 1 and 4 affecting only time window [1, 4]. Hence, the smaller |p′ − p′′ | the more time windows are affected. This effect can not be obtained for periods at the end of the tournament. Consider for example s = 3 solely affecting [4, 7]. Therefore, we extend the set of periods P ≥2 by a set of artificial periods P a : P a = {|P | + 1, . . . , |P | + |S| − 2} P ′ = P ≥2 ∪ P a

(3.16) (3.17)

Note that P a = ∅ if |S| = 2. Now, we consider a time window [p − |S| + 1, p] of length |S| corresponding to each period p ∈ P ′ . Then, in table 3.3 time windows corresponding to periods 7, 8, and 9 are affected by s = 3. We introduce n(n + |S| − 4) integer variables esbi,p , i ∈ T , p ∈ P ′ , representing the surplus of a strength group as opponent of team i in the time window corresponding to period p. Then, we represent the balanced strength groups requirement in n (|S| (n + |S| − 4) + 1) constraints.

Model 3.7: Balanced Strength Groups (SRRTP-IP) min{p,|P |}

p′ =max{p− |S|+1,1}

(xi,j,p′ + xj,i,p′ ) − esbi,p

≤

1

∀ i ∈ T, s ∈ S, p ∈ P ′

j∈s, j =i

(3.18) esbi,p

≤

esbmax

∀i ∈ T

(3.19)

esbi,p

∈

N0

∀ i ∈ T, p ∈ P ′

(3.20)

p∈P ′

Constraint (3.18) sets esbi,p to the surplus of strength group s ∈ S in periods p − |S| + 1 to p. Obviously, esbi,p ∈ {0, . . . , |S| − 1}. Note that artificial periods must be avoided when summing up match variables’ values. Inequality (3.19) restricts the number of violations for each team to no more than a given integer parameter esbmax ∈ N0 .

38


For esbmax = 0 a feasible solution according to team 1 is illustrated in table 3.4. Table 3.4. Opponents’ Strength Groups Balanced period

1

2

3

4

5

6

7

match s

3-1 2

1-5 3

7-1 4

2-1 1

1-4 2

6-1 3

1-8 4

Equally Unchanging Strength Groups Taking into account further constraints the formulation introduced in section 3.2.2 might turn out to be too restrictive. Obviously, we can relax the requirement by increasing escmax . Then, we can not guarantee fairness among teams any more since the number of violations according to two teams might differ. Therefore, we introduce a formulation allowing violations of constraints (3.13) to (3.15) while assuring exactly the same number of violations for each team i ∈ T . We introduce n|S| (n − 2) integer variables esci,p,s , i ∈ T, p ∈ P ≥2 , s ∈ S, counting the number of matches of i against teams belonging to s in periods p − 1 and p. Furthermore c , i ∈ T, p ∈ P ≥2 , s ∈ S, we employ n|S| (n − 2) binary variables fi,p,s indicating a violation of the changing strength groups requirement atc tributed to team i and strength group s in periods p−1 and p (fi,p,s = 1, c fi,p,s = 0 otherwise). Equation (3.21) initializes esci,p,s, i ∈ T, p ∈ P ≥2 , s ∈ S. Consequently, the value of variables esci,p,s is restricted to values out of c {0, 1, 2}. Constraints (3.22) and (3.23) force fi,p,s to be equal to 1 if and only if i has two matches against teams of s in periods p − 1 and p which is a violation of the changing strength group requirement. Constraint (3.24) forces the number of violations to be equal to a given integer parameter ffcix for each team. Instead of this we can use ffcix as an integer variable. Then, the number of violations is not fixed a priori but is obtained as part of the solution. Equally Unbalanced Strength Groups In analogy to “Equally Unchanging Strength Groups” we allow violations according to the balanced strength group concept while assuring


39

Model 3.8: Equally Unchanging Strength Groups (SRRTP-IP) xi,j,(p−1) + xj,i,(p−1) + (3.21) j∈Ss ,j =i

xi,j,p + xj,i,p ) − esci,p,s

=

0

∀ i ∈ T, p ∈ P ≥2 , s ∈ S

c esci,p,s − fi,p,s

≤

1

∀ i ∈ T, p ∈ P ≥2 , s ∈ S (3.22)

c esci,p,s − 2fi,p,s c fi,p,s

≥

0

∀ i ∈ T, p ∈ P ≥2 , s ∈ S (3.23)

=

ffcix

∀i ∈ T

esci,p,s

∈

N0

∀ i ∈ T, p ∈ P ≥2 , s ∈ S (3.25)

c fi,p,s

∈

{0, 1} ∀ i ∈ T, p ∈ P ≥2 , s ∈ S (3.26)

p∈P ≥2

(3.24)

s∈S

exactly the same number of violations for each team i ∈ T . Violations occur as a strength group providing more than one opponent in a specific time window. A violation according to team i, strength group s, and period p directly corresponds to at least one strength group s′ , s′ = s, not providing an opponent of i in the time window corresponding to p. Therefore, we can measure violations vi,p according to team i ∈ T and period p ∈ P ′ by counting the number of strength groups not contained in time window [p − |S| + 1, p]:

vi,p =

s∈S



max

  

min{p,|P |}

1−

p′ =max{p−|S|+1,0} j∈s,j=i

  xi,j,p′ + xj,i,p′ , 0  

We employ n|S| (n + |S| − 4) integer variables esbi,p,s, i ∈ T , p ∈ P ′ , s ∈ S, to count matches between i and teams of s in the time window b , i ∈ T, corresponding to p and n|S| (n + |S| − 4) binary variables fi,p,s p ∈ P ′ , s ∈ S, to indicate violations. Equation (3.27) sets esbi,p,s to the number of matches of team i against teams of strength group s in the time window corresponding b to period p. Constraints (3.28) and (3.29) force fi,p,s to be equal to 1 b b if and only if esi,p,s = 0 holds. Note that s∈S fi,p,s = vi,p . Equation (3.30) assures that the number of violations are equal to ffbix for each team. We have to consider that, obviously, in time windows corresponding to artificial periods |P | + 1 to |P | + |S| − 2 as well as periods 2 to |S|−1 there are strength groups missing due to the number

40


Model 3.9: Equally Unbalanced Strength Groups (SRRTP-IP) min{p,|P |}

(xi,j,p′ + xj,i,p′ )

(3.27)

p′ =max{p− j∈Ss , j =i |S|+1,1}

−esbi,p,s

=

0

∀ i ∈ T, p ∈ P ′ , s ∈ S

b esbi,p,s + fi,p,s

≥

1

∀ i ∈ T, p ∈ P ′ , s ∈ S

esbi,p,s |S|

p∈P ′

s=1

+

b |S|fi,p,s

′

(3.28)

≤

|S|

∀ i ∈ T, p ∈ P , s ∈ S

(3.29)

k

=

ffbix

∀i ∈ T

(3.30)

esbi,p,s

∈

N0

∀ i ∈ T, p ∈ P ′ , s ∈ S

|S|−2 b fi,p,s −2

k=1

b fi,p,s

∈

′

{0, 1} ∀ i ∈ T, p ∈ P , s ∈ S

(3.31) (3.32)

of original periods in the time window. The time window corresponding to period 2, for example, can not contain more than two opponents and, therefore, strength groups. Hence, at least |S| − 2 strength groups are missing in this time window. We take care of this effect by decreasing |S|−2 the number of violations by 2 k=1 k = (|S| − 2)(|S| − 1). In analogy to “Equally Unchanging Strength Groups” we can substitute ffbix by an integer variable so the number of violations is not fixed but is forced to be identical for all teams. 3.2.3 Teams’ Preferences Often teams have preferences to play at home or away in specific periods. In real world sports leagues there are lots of reasons for that. There might be a major regional event so the team located in the region wants to play at home in order to attract more visitors. A prominent example is Bayern Munich preferring to play at home during the Octoberfest. The other way round teams might prefer to play away during major events in order to disburden the region’s infrastructure. Furthermore, teams might prefer to play away due to construction works at the team’s stadium causing lowered seating capacity. Last but not least some teams like to have the season’s final match at home. In order to represent a team i’s preference in period p we introduce the trivalent parameter pri,p . If team i prefers to play at home in period p we set pri,p to 1, if it likes to play away pri,p is set to −1; if team i has


41

no preference at all we let pri,p = 0. Obviously, we require a reasonable mechanism to handle teams having different numbers of preferences. We do not limit the number of preferences pri = p∈P |pri,p | of team i ∈ T . Given a number cp of preferences to be conceded we construct sports league schedules allowing exactly min{cp, pri } preferences for each team i. We employ n(n − 1) binary variables npi,p being equal to 1 if and only a preference of team i in period p is neglected.

Model 3.10: Teams’ Preferences (SRRTP-IP) xi,j,p − pri,p xj,i,p + 2npi,p = |pri,p | ∀ i ∈ T, p ∈ P pri,p j∈T,j =i

j∈T,j =i

(3.33)

npi,p − max {pri − cp, 0}

=

0

∀i ∈ T

npi,p

∈

{0, 1} ∀ i ∈ T, p ∈ P (3.35)

(3.34)

p∈P

Equation (3.33) sets npi,p to 1 if and only if team i’s preference in period p is neglected. In order to line (3.33) out in detail we distinguish three cases. First, if team i has a preference to play at home in period p (pri,p = 1) the right hand side equals 1 and the first two terms of the left hand side sum up to 1 or -1 if team i plays at home or away, respectively, in period p. Consequently, npi,p = 1 if and only if team i’s preference in period p is neglected. Second, if team i has a preference to play away in period p (pri,p = −1) the right hand side is equal to 1, again, and the first two terms of the left hand side sum up to 1 or -1 if team i plays away or at home, respectively, in period p. Again, npi,p = 1 if and only if team i’s preference in period p is neglected. Third, if team i has no preference in period p the right hand side equals 0 and the first two terms of the left hand side sum up to 0 no matter where i plays in period p. Hence, npi,p = 0. Equation (3.34) forces the number of neglected preferences for each team i to be equal to the number of preferences given by i minus the number of preferences to be obeyed. Obviously, restriction (3.34) can be formulated as the number of neglected preferences for each team to be no more than the maximum expression in order to guarantee a certain number of obeyed preferences. Furthermore, we can think of cp as integer variable. Then,

42


no number of preferences to concede is given in advance but determined as part of the solution.

3.3 Computational Study 3.3.1 Generating Problem Instances In order to establish benchmarks for algorithms tackling various sports league scheduling problems we provide generation schemes for problem instances covering all issues introduced in sections 3.1 and 3.2 based on the structure of single RRTs and mirrored double RRTs, respectively. Moreover, we propose key figures enabling us to manipulate the instances’ characteristics. Problem Size: The size of an instance is determined by a number n of teams which is given as parameter for the generation process. Recall that n can be restricted to be even w.l.o.g. as shown in section 1.3. HAP sets: In order to consider the HAP set-based RRT problem (see section 2.4.2) we generate random HAP sets as precondition to the scheduling process. We randomly generate them by selecting n2 teams to be teams playing at home for each period p ∈ P . This implies the remaining teams to play away in p. Note that this does not guarantee feasibility of the HAP set. No parameter is necessary to guide the generation of HAP sets. Forbidden Matches: If the set of possible matches is restricted according to section 3.1.1 we need a parameter πi,j,p for each match indicating whether it is forbidden (πi,j,p = 0) or not (πi,j,p = 1). We control the fraction of matches which are allowed by parameter Pπ . Pπ gives the probability for each match (i, j, p) to be possible. When constructing an instance each π is randomly chosen from {0, 1} acP Pi,j,p P πi,j,p

\{i} p∈P ≈ Pπ . cording to Pπ leading to i∈T j∈T n(n−1)2 Stadium Availability: As described in section 3.1.1 stadium unavailability can be seen as a set of forbidden matches. However, a special case, namely πi,j,p = 0 for each j ∈ T \ {i}, is necessary to represent i’s stadium to be unavailable in period p. Therefore, we introduce a distinct parameter PπS for guiding the number of times a stadium is not available. Parameter PπS gives the probability for a team i’s stadium to be available in a specific period p. The availability of each team’s stadium in P each P period is chosen randomly from {0, 1} according to PπS

max

{πi,j,p }

j∈T \{i} ≈ PπS . leading to i∈T p∈P n(n−1) Regions: If regions’ capacities are considered we have to specify the set of regions R according to section 3.1.2. Each region is defined

3.3 Computational Study

43

by both, the teams it contains and the maximum number of matches being allowed in this region per period, explicitly. Highly Attended Matches: We propose to guide characteristics of instances considering highly attended matches employing two parameters. First, Pa gives the probability for a match between teams i ∈ T and j ∈ T \ {i} at i’s home to be highly attended. Then, parameter a is chosen randomly from {0, 1} according to Pa resulting P Pi,j ai,j

j∈T \{i} in i∈T n(n−1) ≈ PaS . Second, the maximum number of attractive matches per period amax is given explicitly. Reasonably, values for amax are from {1, . . . , n2 − 1}. Parameter amax being equal to 0 does not allow any attractive match at all while amax being no less than n2 does not define any restriction at all since there are exactly n2 matches per period. Breaks: If breaks are considered we distinguish four classes of instances. First, the overall goal must be chosen to be either minimization of number of breaks or arranging exactly one break per team. Second, we allow and forbid, respectively, breaks to occur in the second period. Beside these choices there are no further parameters necessary. Opponents’ Strengths: In order to consider opponents’ strengths three parameters must be defined: First, the mode of consideration must be determined. It can be chosen from “changing”, “balanced”, “equally unchanging”, and “equally unbalanced”. Second, the set of strength groups has to be defined. Following the concept outlined in section 3.2.2 the set of strength groups can be defined by specifying the number of strength groups |S| explicitly. Third, we have to specify the number of violations.

3.3.2 Computational Results We line out several key figures in order to judge the computational burden caused by the single RRT problem and specific additional requirements. First of all, we state average run times in seconds (“r.t.”). Run times are based on those instances which could be solved to optimality. Moreover, number of test instances (“i.”) and the number of instances an optimal solution could be found for (“s.f.”) as well as the number of problems having no solution at all (“n.s.”) were contraposed. Note that runs not leading to an optimal solution or to detection of instance’s infeasibility have been aborted because of lack of memory.

44


Basic Problem First, we observe the run time behavior when solving the single RRT problem without any additional restrictions. Results are provided in table 3.5. Table 3.5. Comp. Results for Single RRT Problem n i. s.f. 6 8 10 12 14 16 18

20 20 20 20 20 20 3

r.t.

20 0.01 20 0.05 20 0.37 20 2.03 20 34.43 20 2811.18 1 74601.00

We clearly observe an explosion of run times as problem sizes grow. While run times for problems with up to 12 teams are rarely worth mentioning they extremely raise for larger instances. Instances with 16 teams can be solved to optimality in 47 minutes on average but only one instance with 18 teams could be solved to optimality. This took more than 20.5 hours. Cplex runs out of memory while solving the remaining instances having 18 teams. Figure 3.1 illustrates the exponential run time behavior. Note that the logarithm ln(t) of run time t is shown. The function progresses in an almost linear manner which corresponds to the proof given in section 2.1. The last section is drawn as dashed line since there was only one instance indicating this run time. Nevertheless, this time fits into the linear scheme recognized for smaller instances. Those runs suffering from lack of memory were aborted after about 14 hours. Forbidden Matches As described in section 3.1.1 forbidden matches cut down solution space since solutions must not contain a forbidden match. This might lead to decreasing run times. On the other hand finding feasible solutions gets more difficult and cost oriented branching mechanisms might get stuck in infeasible paths more likely. Hence, we have two reverse effects influencing run times in comparison to those presented for the basic


45

12.0 10.0 8.0 6.0 ln(t)

4.0 2.0 0.0

−2.0 −4.0 −6.0 6

8

10

12 n

14

16

18

Fig. 3.1. Run time behavior for single RRT problem

problem. In order to study the impact of those effects we tested instances with 6 to 18 teams and probabilities for matches to be allowed of Pπ ∈ {0.9, 0.8, 0.7, 0.6, 0.5}. Results are lined out in table 3.6. For each class of instances we give size, number of instances, number of instances solved to optimality, number of infeasible instances, and run times, respectively. Up to 10 teams we observe almost equal run times not different from those determined for the single RRT problem for all Pπ . For larger instances run times do not provide a clear idea about the influence of the number of teams and Pπ on run times. For example while Pπ = 0.9 leads to largest run times for instances having 14 teams it leads to the smallest run times for instances having 16 teams. Unfortunately, we can not observe any systematic correlation here. However, characteristics of average run times’ behavior for instances with forbidden matches do not severely differ from those presented for the single RRT problem shown in figure 3.1, therefore, we restrain to give them here explicitly. We emphasize that infeasibility of instances is rather odd even if about 50% of matches are forbidden. For a given Pπ instances seem to be the more vulnerable to be infeasible the smaller they are. Infeasibility was detected for only a few of the smallest instances.

46

3 Real World Problems Table 3.6. Comp. Results for Forbidden Matches Pπ =0.9 Pπ =0.8 n i./s.f./n.s. r.t. i./s.f./n.s. 6 8 10 12 14 16 18

20/20/0 0.01 20/20/0 0.05 20/20/0 0.39 20/20/0 1.84 20/20/0 44.16 20/20/0 2236.30 3/0/0 ———

20/20/0 0.01 20/20/0 0.03 20/20/0 0.35 20/20/0 2.30 20/20/0 32.95 20/20/0 2252.25 3/1/0 18166.90

Pπ =0.6 n i./s.f./n.s. 6 8 10 12 14 16 18

Pπ =0.7 r.t. i./s.f./n.s.

20/20/0 0.01 20/20/0 0.03 20/20/0 0.32 20/20/0 1.80 20/20/0 34.53 20/20/0 2647.91 3/2/0 43820.15

Pπ =0.5 r.t. i./s.f./n.s.

20/19/1 0.01 20/20/0 0.04 20/20/0 0.42 20/20/0 2.10 20/20/0 41.41 20/20/0 3898.66 3/1/0 27288.90

r.t.

r.t.

20/17/3 0.01 20/20/0 0.04 20/20/0 0.40 20/20/0 2.03 20/20/0 30.15 20/20/0 3094.34 3/1/0 29698.80

Stadium Availability As outlined in section 3.1.1 we represent stadiums unavailability by forbidden matches. In addition to the results for forbidden matches we have a further look on run times when stadium availability is considered because of the high relevance in real world RRTs. We create test instances having 6 to 18 teams and PπS ∈ {0.9, 0.8, 0.7}. Results are given in table 3.7. Trivially, probability of instances’ infeasibility is higher if probability of stadium availability is lower. This thought is confirmed by the fraction of infeasible instances of each problem class. No instance having PπS = 0.9 is infeasible while for each problem size there are infeasible instances having PπS = 0.7. Comparing tables 3.6 and 3.7 we conclude that probability of instance’s infeasibility is clearly higher when considering stadium availability instead of arbitrarily forbidden matches even if PπS = Pπ . By contrast decreasing probability of stadium availability lowers run time requirements. This can be observed for n < 18. Note that we can think of the single RRT problem as a problem having PπS = 1. Then,


47

Table 3.7. Comp. Results for Stadium Availability PπS =0.9 n i./s.f./n.s. 6 8 10 12 14 16 18

PπS =0.8 r.t. i./s.f./n.s.

10/10/0 0.01 10/10/0 0.03 10/10/0 0.43 10/10/0 2.33 10/10/0 31.31 10/10/0 2100.76 3/1/0 64987.30

PπS =0.7 r.t. i./s.f./n.s.

10/10/0 0.01 10/10/0 0.03 10/9/1 0.36 10/10/0 2.02 10/10/0 27.48 10/9/1 1308.86 3/2/0 29126.90

r.t.

10/8/2 0.01 10/6/4 0.03 10/5/5 0.17 10/7/3 1.42 10/9/1 9.71 10/5/5 1085.94 3/3/0 30339.27

results from table 3.5 fit into this observation since run times according to the single RRT problem are larger than those in table 3.7 for n > 12. For n = 18 run times may not be representative since we could only test very few instances. However, three out of three instances with PπS = 0.7 and n = 18 could be solved to optimality. The fraction of instances the solution process terminated for (either finding a feasible solution or detecting infeasibility) decreases with increasing PπS which may be an indicator for larger run time requirements. Furthermore, considering stadium availability seems to lead to lower run times than general forbidden matches do. Although instances considering stadium availability can be considered special cases for general forbidden matches run times for are significantly lower than for 12 < n < 18 and PπS = Pπ . HAP–set–based single RRT Problem Scheduling a single RRT to minimum cost with respect to a given HAP set can be seen as a special case of taking care of stadium availability with PπS = 0.5. However, due to the instance generation scheme infeasibility of an instance with given HAP set is less probable than infeasibility of an instance with PπS = 0.7 (see above) and, therefore, presumably less probable than infeasibility of an instance with PπS = 0.5. This is clearly confirmed in table 3.8: only one out of 70 randomly generated instances is infeasible while 20 out of 63 instances generated with PπS = 0.7 are infeasible according to table 3.7. Run times are significantly lower than those for instances considering stadium availability given in table 3.7 and, consequently, than those for instances incorporating general forbidden matches and for the single RRT problem. Since this section is the one considering the restrictions

48

3 Real World Problems Table 3.8. Comp. Results for HAP–set–based Single RRT Problem n i./s.f./n.s. 6 8 10 12 14 16 18

r.t.

10/10/0 0.01 10/9/1 0.01 10/10/0 0.18 10/10/0 0.62 10/10/0 3.28 10/10/0 72.73 10/10/0 5987.32

proofed to be most run time reducing in the course of the previous sections we illustrate the run time behavior in figure 3.2. 10.0 8.0 6.0 ln(t)

4.0 2.0 0.0

−2.0 −4.0 −6.0 6

8

10

12 n

14

16

18

Fig. 3.2. Run time behavior for HAP–set–based single RRT problem

In spite of the smaller run times we, again, observe a clearly exponential run time behavior which corresponds to the proof given in section 2.4.2.


49

Highly Attended Matches When considering limitation of the number of highly attended matches per period we expect the same reverse effects as lined out before: reduction of solution space versus difficulty of finding feasible solutions. We created instances with probability of Pa = 0.2 for a match to be attractive. We vary amax depending on the instances’ size n. Note that for n = 10 and Pa = 0.2 there is one attractive match per period on average. Therefore, we increase amax from 1 for n = 8 to 2 for n = 10. Table 3.9. Comp. Results for Highly Attended Matches n Pa amax i./s.f./n.s. 6 8 10 12 14 16 18

0.2 0.2 0.2 0.2 0.2 0.2 0.2

1 1 2 2 2 2 2

r.t.

10/10/0 0.01 10/10/0 0.09 10/10/0 0.29 10/10/0 2.78 10/10/0 96.90 10/10/0 16925.70 3/0/0 —

Obviously, amax is differently restrictive for instances having 6 and 8 teams and having 10, 12, 14, 16, and 18 teams, respectively. This leads to a distortion of run times. However, run times in table 3.9 are comparable to those of the single RRT problem in table 3.5. We clearly observe larger run times than for the single RRT problem if considering highly attended matches. Regions’ Capacity As outlined in section 3.1.2 we can think of a huge amount of constellations taking regions into account. For a given size of instances n there can be various numbers of regions, various sizes of regions, and various assignments of teams to regions (which can influence run times due to different costs assigned to different teams). We exemplarily create three classes R2 , R3 , and R4 of instances. In each class we have a given number |R| = 2, |R| = 3, and |R| = 4 of regions. The concept is to let regions have sizes as close to each other as possible without being identical for all regions. In R2 there are two disjoint regions of size n2 − 1 and n2 + 1, respectively. In R3 regions

50


n are disjoint having minimum size of 3 . n mod 3 teams have size

n there are four disjoint regions having minimum size of + 1. In R 4

n3

n 4 . n mod 4 teams have size 4 + 1. ′ The number ′ region R ⊂ T is limited by of matches in a specific |R | |R′ | + 1. Since CR′ ≥ must hold (see section 3.1.2) CR′ = 2 2 limitation for each region is quite restrictive. Table 3.10. Comp. Results for Regions |R| = 2 n |R′ | CR′ i./s.f. 6 8 10 12 14 16 18

2/4 3/5 4/6 5/7 6/8 7/9 8/10

2/3 2/3 3/4 3/4 4/5 4/5 5/6

n 6 8 10 12 14 16 18

r.t. |R′ |

10/10 0.01 10/10 0.09 10/10 0.40 10/10 3.65 10/10 81.35 10/10 4080.78 3/0 —

|R′ |

|R| = 3 CR′ i./s.f.

r.t.

2/2/2 2/2/2 10/10 0.01 2/3/3 2/2/2 10/10 0.08 3/3/4 2/2/3 10/10 0.83 4/4/4 3/3/3 10/10 4.82 4/5/5 3/3/3 10/10 345.28 5/5/6 3/3/4 10/5 32663.66 6/6/6 4/4/4 3/0 —

|R| = 4 CR′ i./s.f.

r.t.

1/1/2/2 1/1/2/2 10/10 0.02 2/2/2/2 2/2/2/2 10/10 0.03 2/2/3/3 2/2/2/2 10/10 0.83 3/3/3/3 2/2/2/2 10/10 16.21 3/3/4/4 2/2/3/3 10/10 546.06 4/4/4/4 3/3/3/3 10/10 20775.20 4/4/5/5 3/3/3/3 3/0 —

Results are given in table 3.10. Each single class of problem instances proofed to be feasible. Almost all instances could be solved to optimality. No instance having 18 teams could be solved to optimality. However, for those instances not solved to optimality feasible solutions were found. With respect to the run times we can conclude that mostly a larger number of regions means higher run times. We observe exceptions from this rule for instances having 16 teams and 3 and 4 regions, respectively. A possible explanation for this effect is that limitation for the number of matches in a specific region is much more restrictive in those instances


51

having 3 regions. For example for instances having 16 teams and 3 regions the number of matches per region is restricted to about 0.62 times the number of teams in a region on average. This rate amounts to 0.75 for instances with 16 teams and 4 regions and is, therefore, less restrictive. Furthermore, this effect elucidates why only half of the instances having 16 teams and 3 regions could be solved to optimality. The basic problem can be seen as a problem having |R| = 1, R′ = T , and CR′ = n2 + 1 according to the concept outlined above. Then, it fits into our observation that run times are lower if the number of regions is smaller. Breaks We study problems incorporating requirements considering breaks as introduced in section 3.2.1. In detail we solve problems requiring a minimum number of breaks (allowing and forbidding breaks in the second period) and having exactly one break per team. Run times are lined out in table 3.11. Table 3.11. Comp. Results for Breaks n 6 8 10

min no min no, ex. one b. ex. one b., not 2nd not 2nd 0.23 26.79 —

0.21 26.63 —

0.25 38.44 —

— 74.12 —

Clearly, allowing and forbidding breaks in the second period has no impact on run times when a minimum number of breaks is required. For n = 10 Cplex aborted the solution process due to lack of memory after about 15 hours of run time for each single instance. In most cases not even a single feasible solution was found. This, first of all, shows the enormous increase of run time for solving these problems to optimality. Second, it gives an idea of the difficulty to find even one feasible solution when cost oriented branching is employed instead of the standard generation scheme mentioned in section 3.2.1. This observation motivates the development of a branching scheme specialized in single RRTs having the minimum number of breaks in chapter 5. Run times for instances requiring exactly one break per team are even higher. Again, we can not solve instances with more than n =

52


8 teams. Here, allowing and forbidding breaks in the second period influences run times. Average run time forbidding breaks in the second period is about twice as high as run times allowing them. This coincides with computational effort to find feasible solutions at all. Run time to find the first feasible solution forbidding breaks in the second period is about three times as high as run times when we allow them. Opponents’ Strengths In section 3.2.2 four variants to consider a team’s opponents’ strengths in order to establish fairness are proposed. We tested instances having up to n = 18 teams for all variants. The number of strength groups is set to 2 and n2 , respectively. For “changing” and “balanced” we additionally set the number of violations escmax and esbmax , respectively, to 0. Considering “equally unchanging” and “equally unbalanced” we additionally set the number of violations ffcix and ffbix to 1, respectively, Therefore, each team must violate the opponent strength constraint exactly once. Table 3.12. Comp. Results for Changing Opponents’ Strength Groups |S| = 2 |S| = n i./s.f./n.s. r.t. i./s.f./n.s. 6 8 10 12 14 16 18

10/0/10 — 10/10/0 0.02 10/0/10 — 10/10/0 0.18 10/0/10 — 10/10/0 10.33 10/0/10 —

n 2

r.t.

10/0/10 — 10/10/0 0.34 10/10/0 4.59 10/10/0 233.64 10/10/0 11559.40 3/0/0 — — —

In tables 3.12 to 3.15 results for all classes of instances are provided. Referring to the question of instances’ feasibility posed in section 3.2.2 we can identify problem classes considering changing opponents’ strengths being infeasible due to values of n and |S| according to table 3.12. For n ≤ 18, n mod 4 = 2, and |S| = 2 we observe infeasibility. We conjecture this to be true for n > 18. Furthermore, n = 6 and |S| = 3 leads to infeasibility as well. Run times for feasible instances having |S| = 2 are significantly lower than for the single RRT problem of corresponding size n. To the


53

contrary run times are significantly higher for |S| = n2 in comparison to both, the single RRT problem and instances with changing opponents’ strengths and |S| = 2. Consequently, instances having 16 teams and more can not be solved to optimality. Table 3.13. Comp. Results for Balanced Opponents’ Strengths |S| = 2 |S| = n2 n i./s.f./n.s. r.t. i./s.f./n.s. r.t. 6 8 10 12 14 16 18

10/0/10 — 10/0/10 — 10/10/0 0.03 10/10/0 0.03 10/0/10 — 10/0/10 — 10/10/0 0.61 10/10/0 0.50 10/0/10 — 10/0/10 — 10/10/0 10.09 10/0/10 9.26 10/0/10 — 10/0/10 —

Inspecting table 3.13 again we identify problem classes being infeasible. Note that balanced opponents’ strengths structure is a special case of changing opponents’ strengths structure. Therefore, it is straightforward that n and |S| is infeasible according to balanced opponent’s strengths if n and |S| is infeasible according to changing opponents’ strengths. However, we identify some instance classes being feasible according to changing opponent’s strengths but being infeasible according to balanced opponents’ strengths: For n ≤ 18, n mod 4 = 2, and |S| = n2 we observe infeasibility. Again, we conjecture this be valid for n > 18. Run times for both |S| = 2 and |S| = n2 are significantly lower than for the single RRT problem. Run times for |S| = n2 are slightly lower than those for |S| = 2 which is in contrast to the relation observed for changing opponents’ strengths. Note that instances with |S| = 2 and changing opponents’ strengths are balanced as well with |S| = 2. Hence, results according to |S| = 2 in tables 3.12 and 3.13 only differ slightly and due to cost structure and model formulation, respectively. Run times for equally unchanging opponents’ strengths are given in table 3.14. Here we observe remarkably higher run times compared with changing opponents’ strengths given in table 3.12. The reason for this might be the larger number of binary variables necessary to represent the equally unchanging opponents’ strengths requirement. Furthermore, integer variables not restricted to binary values are incorpo-

54


Table 3.14. Comp. Results for Equally Unchanging Opponents’ Strengths |S| = 2 |S| = r.t. i./s.f./n.s. n i./s.f./n.s. 6 8 10 12 14

n 2

r.t.

10/0/10 — 10/0/10 — 10/10/0 0.76 10/10/0 8.26 10/0/10 — 10/10/0 3116.31 10/10/0 195.01 3/0/0 — 3/0/0 — — —

rated. As far as table 3.14 provides insights into this topic exactly the same classes of problem instances seem to be infeasible as for changing opponents’ strengths. Finally, run times for equally unbalanced opponents’ strengths are given in table 3.15. Table 3.15. Comp. Results for Equally Unbalanced Opponents’ Strengths |S| = 2 |S| = n i./s.f./n.s. r.t. i./s.f./n.s. 6 8 10 12 14

10/0/10 — 10/10/0 0.63 10/0/10 — 10/10/0 639.20 3/0/0 —

n 2

r.t.

10/0/10 — 10/10/0 2.40 10/0/10 — 10/10/0 17290.30 3/0/0 —

Run times are clearly higher than for balanced opponents’ strengths and equally unchanging opponents’ strengths given in tables 3.13 and 3.14, respectively. Again, results give only a slight idea of infeasible instance classes. However, as it was the case for changing and equally unchanging opponents’ strengths there seems to be no difference according to problems feasibility between balanced and equally unbalanced opponents’ strengths. Opponents’ Strengths and Breaks We line out computational results for a combination of strength group requirements as well as break requirements as introduced in sections 3.2.1 and 3.2.2. In contrast to other results corresponding to IP model


55

techniques in the chapter at hand we combine two requirements, here, because they have a special meaning in context of the branch-and-price (B&P) approach developed in chapter 6. Exemplarily we postulate the minimum number of breaks allowing breaks in the second period and changing opponents’ strengths considering 2 and n2 strength groups, respectively. Table 3.16. Comp. Results for Breaks and Changing Opponents’ Strengths |S| = 2 |S| = n2 n i./s.f./n.s. r.t. i./s.f./n.s. r.t. 8 10/10/0 3.71 10/10/0 83.81 10 — — 3/0/0 — — — — 12 3/0/0

Clearly, instances having 6 teams or 10 teams and 2 strength groups have no solution at all, see table 3.16. Furthermore, no instance having more than 8 teams could be solved to optimality. Solution process was aborted due to lack of memory instead. Run times for 8 teams and n2 = 4 strength groups are substantially larger than run times for both minimum number of breaks in table 3.11 and changing strength groups in table 3.12. In analogy, considering 2 strength groups for 8 teams (and the minimum number of breaks) run times are larger than for the problem considering only changing strength groups. However, they are clearly smaller than for problems considering only the minimum number breaks. This observations backs up the one in section 3.3.2 stating that balanced opponents’ strengths (changing opponents’ strengths with |S| = 2 means balanced opponents’ strengths) reduces run times. Teams’ Preferences We consider six classes of instances considering teams’ preferences as introduced in section 3.2.3: teams specify 1-2, 1-3, and 2-4 preferences, and exactly or at least 1, 1, and 2 preferences, respectively, have to be considered. Obviously, one can think of more preferences if the number of teams (and, therefore, the number of periods) is larger. For the sake of comparability we refuse to do so. Results are given in table 3.17. For each single problem class and size we can conclude that run times are significantly higher than for the single RRT problem having

56

3 Real World Problems Table 3.17. Comp. Results for Teams’ Preferences 1-2, exactly 1 1-3, exactly 1 2-4, exactly 2 n i./s.f./n.s. r.t. i./s.f./n.s. r.t. i./s.f./n.s. r.t. 6 8 10 12 14 16 18

10/10/0 0.01 10/10/0 0.09 10/10/0 0.50 10/10/0 3.14 10/10/0 66.05 10/10/0 6192.64 3/1/0 156683.00

10/10/0 0.02 10/10/0 0.14 10/10/0 0.86 10/10/0 4.01 10/10/0 157.36 10/10/0 12223.80 3/0/0 —

10/10/0 0.02 10/10/0 0.20 10/10/0 1.20 10/10/0 5.84 10/10/0 359.98 10/10/0 21002.60 3/0/0 —

1-2, at least 1 1-3, at least 1 2-4, at least 2 n i./s.f./n.s. r.t. i./s.f./n.s. r.t. i./s.f./n.s. r.t. 6 8 10 12 14 16 18

10/10/0 10/10/0 10/10/0 10/10/0 10/10/0 10/10/0 3/0/0

0.01 0.05 0.39 2.81 90.45 3596.75 —

10/10/0 10/10/0 10/10/0 10/10/0 10/10/0 10/10/0 3/0/0

0.02 0.06 0.48 3.05 67.22 2841.99 —

10/10/0 0.01 10/10/0 0.09 10/10/0 0.66 10/10/0 2.90 10/10/0 53.91 10/10/0 13205.00 3/0/0 —

identical size. Furthermore, run times for each problem class considering a given exact number of preferences to be granted are higher than for the corresponding class considering a given minimum number of preferences to be granted. This effect might result from the difficulty to find feasible solutions. Obviously, the set of solutions having an exact number p′ of granted preferences per team is a subset of the set of solutions having p′ as a minimum number of preferences to be granted per team. As we can see increasing the number of preferences and increasing the number of preferences to be granted leads to higher run times if an exact number of preferences to be fulfilled is given. Again, the reason for this probably is the difficulty to find feasible solutions: increasing the number of given preferences implies a rising number of preferences which must be neglected. This effect does not come into play if we consider a minimum number of preferences to be fulfilled. Instead, by increasing the number of preferences there is more freedom to choose the number of preferences to be fulfilled. Therefore, there is a tendency

3.4 Summary

57

that run times are lower if 1 to 3 preferences are given in comparison with 1 to 2 preferences.

3.4 Summary In this chapter we pick up several prominent real world requirements in the context of RRT scheduling. Furthermore, we substantiate requirements related to fairness which mostly have been proposed in abstract terms in literature so far. We formally define the requirements by means of IP modelling techniques. Moreover, we study the run time behavior resulting from the optimization models using Cplex. We observe exponential run time behavior for nearly all variations of the single RRT problem. Therefore, run times are exorbitant as soon as problem sizes grow relevant for real world problems except. We detect a single exception from this rule: If we consider balanced opponents’ strength groups run times remain manageable. Hence, these variants might serve as basis for real world problems.

4 Combinatorial Properties of Strength Groups

In this chapter we provide insights into combinatorial aspects concerning strength groups in RRTs as presented in Briskorn [7]. Note that indices are “0-based” throughout this chapter. We refer to section 3.2.2 for a motivation of strength groups and basic definitions. Definition 4.1. A single RRT where no team plays against teams of the same strength group in two consecutive periods is called groupchanging. Definition 4.2. A single RRT where no team plays more than once against teams of the same strength group within |S| consecutive periods is called group-balanced. Group-changing single RRTs and group-balanced single RRTs correspond to concepts presented in section 3.2.2. An interesting question is how n and |S| can be chosen such that a group-changing single RRT or a group-balanced single RRT exists. We empirically proof specific values of n and |S| to be infeasible in section 3.3.2. Moreover, we conjecture classes of values to be infeasible in general. These conjectures are proven to be true in the chapter at hand. Furthermore, we proof complexity of cost minimization problems being in line with problems considered in chapter 2 and considering strength groups in section 4.4. For the sake of convenience we introduce some short notations. A specific strength group is referred to as Sk ∈ S with k ∈ {0, . . . , |S|−1}. We denote the index of the strength group of team i ∈ T by S(i) ∈ S, i.e. i ∈ SS(i) . The opponent of team i ∈ T in period p ∈ P is denoted by oi,p ∈ T .

60


4.1 Factorizations First, we focus on graph theoretical aspects. A r-factor of a graph G = (V, E) is a set of edges E ′ ⊆ E such that each node i ∈ V is incident to exactly r edges in E ′ . A r-factorization of G is a partition of its edges into r-factors. For details we refer the reader to Wallis [85], for example. A near-1-factor of G is a set of edges E ′ ⊆ E such that each node but one is incident to exactly one e ∈ E ′ . This node is incident to no e ∈ E ′ . A near-1-factorization of G is a partition of its edges into near-1-factors. An ordered r-factorization (near-1-factorization) is a r-factorization (near-1-factorization) where the r-factors (near-1-factors) are ordered. 4.1.1 Ordered 1-Factorization of Kk,k The complete balanced bipartite graph Kk,k , k ∈ N, is well known to have an ordered 1-factorization F bip , as proposed for example in de Werra [19]. Let the color classes (see Schrijver [78]) of Kk,k be defined by V0 := {i | i ∈ {0, . . . , k − 1}} and V1 := {i | i ∈ {k, . . . , 2k − 1}}. Then, bip , where F bip = F0bip , . . . , Fk−1

Flbip = {[m, k + (m + l)mod k] | m ∈ {0, . . . , k − 1}} ∀ l ∈ {0, . . . , k − 1} . Here, [i, j], i, j ∈ V , denotes the edge incident to i and j. Note that differences i − j and j − i are not equal to 1 in F0bip unless k = 1. An example with k = 4 is given 4.1. in figure

We emphasize that in bip k F k no edge [m, k + n], m, n ∈ 0, . . . , 2 − 1 , and no edge [m, k + n], 2

m, n ∈ k2 , . . . , k − 1 , is contained if k is even. 4.1.2 Ordered 1-Factorizations of Kk

It is well known that there is an ordered 1-factorization consisting of k − 1 1-factors of each Kk , k even. The canonical 1-factorization is defined in the following (all indices being taken modulo k − 1):

4.1 Factorizations

61

0

4

0

4

0

4

0

4

1

5

1

5

1

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1

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2

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2

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2

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2

6

3

7

3

7

3

7

3

7

F0bip

F1bip

F2bip

F3bip

Fig. 4.1. 1-Factorization of K4,4

c , where F c = F0c , . . . , Fk−2 k c Fl = [l, k − 1] ∪ [l − m, l + m] | m ∈ 1, . . . , − 1 2 ∀ l ∈ {0, . . . , k − 2}. If not stated otherwise we refer to F c as 1-factorization of Kk in the remainder. Note that we can force F kc −1 to exclusively contain edges of 2

form [i, i + 1], i even, by a simple mapping σ : V → V . An illustration of the canonical 1-factorization of K6 is given in figure 4.2.

2

3

2

3

5 1

1

1

2

3

5

1

4

1

4 0

0 F2c

3

5 4

0 F1c

2

5 4

0 F0c

3

5 4

0

2

F3c

F4c

Fig. 4.2. Canonical 1-Factorization of K6

If k is odd we can construct a near-1-factorization nF c consisting of k near-1-factors by simply letting the node matched with k in F c of Kk+1 be unmatched in each Flc , l ∈ {0, . . . , k − 1}:

62


c nF c = aF0c , . . . , aFk−1 , where k c nFl = [l − m, l + m] | m ∈ 1, . . . , − 1 ∀ l ∈ {0, . . . , k − 1} . 2 Since F c and nF c is a so called started induced 1-factorization and near-1-factorization, respectively, each number in {1, . . . , 2k − 2} can be found as a difference i−j or j−i of an edge [i, j] in each 1-factor in F c and nF c . Next, we introduce 1-factorizations and near-1-factorizations for Kk , k ∈ N, k > 3, where not each of those numbers is contained in each 1-factor. The binary 1-factorization as proposed in de Werra [19] can be constructed for K2k if k even. Let V0 := {i | i ∈ {0, . . . , k − 1}} and V1 := {i | i ∈ {k, . . . , 2k − 1}} be a partition of V . Then, 1-factor Flb,e is set to Flbip as introduced in section 4.1.1 for each l ∈ {0, . . . , k − 1}. Hence, each edge between V0 and V1 is contained in 1-factors F0b,e b,e b,e to Fk−1 . Additionally, 1-factors Fkb,e to F2k−2 are constructed as 1factorization according to V0 and V1 , respectively. Then, 1-factorization F b,e is defined as follows: b,e , where F b,e = F0b,e , . . . , F2k−2

Flb,e = {[m, k + (m + l)mod k] | m ∈ {0, . . . , k − 1}} ∀ l ∈ {0, . . . , k − 1} , Flb,e = [l, k − 1] ∪ [l + k, 2k − 1] ∪ {[(l − k − m)mod k, (l − k + m)mod k] | m ∈ {0, . . . , k − 1}} ∪ {[k + (l − k − m)mod k, k + (l − k + m)mod k] | m ∈ {0, . . . , k − 1}} ∀ l ∈ {k, . . . , 2k − 2} .

Since F0b,e is based on F0bip none of the differences i − j or j − i is equal to 1 if k > 1. Figure 4.3 represents the binary 1-factorization of K8 . We extend the binary 1-factorization of K2k to the case where k b,o bip is odd. 1-factors F0b,o to Fk−2 are defined as 1-factors F0bip to Fk−2 b,o b,o according to V0 and V1 . 1-factors Fk−1 to F2k−2 are composed of near1-factors according to V0 and V1 . Note that one node of both, V0 and V1 , is unmatched. These nodes are matched and form the edges between b,o V0 and V1 missing from F0b,o to Fk−2 .

4.1 Factorizations

63

0

4

0

4

0

4

0

4

1

5

1

5

1

5

1

5

2

6

2

6

2

6

2

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3

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3

7

3

7

3

7

F0b,e

F1b,e

F2b,e

F3b,e

0

2 4

6

0

2 4

6

0

2 4

6

1

3 5

7

1

3 5

7

1

3 5

7

F4b,e

F5b,e

F6b,e

Fig. 4.3. Binary 1-Factorization of K8

b,o , where F b,o = F0b,o , . . . , F2k−2

Flb,o = {[m, k + (m + l)mod k] | m ∈ {0, . . . , k − 1}} ∀ l ∈ {0, . . . , k − 2} , Flb,o = [l − k + 1, k + (l − k)mod k] ∪ {[(l − k + 1 − m)mod k, (l − k + 1 + m)mod k] | m ∈ {0, . . . , k − 1}} ∪ {[k + (l − k − m)mod k, k + (l − k + m)mod k] | m ∈ {0, . . . , k − 1}} ∀ l ∈ {k − 1, . . . , 2k − 2} .

Theorem 4.1. F b,o is a 1-factorization of K2k , k odd. Proof. We show that Flb,o is a 1-factor for each l ∈ {0, . . . , 2k − 2} and that F b,o is a partition of edges of K2k . Obviously, Flb,o is a 1-factor for l ∈ {0, . . . , k − 2} since it is defined by a 1-factor according to F bip considering a partition of V . Flb,o , l ∈ {k − 1, . . . , 2k − 2}, is defined by near-1-factors of both, V0 and V1 .

64


Therefore all but nodes l − k + 1 and k + (l − k)mod(k − 1) are matched implicitly in Flb,o . These two nodes are matched explicitly. Each edge between V0 and V1 but [m, (m − 1) mod (k − 1)], m ∈ {0, . . . , k − 1} is contained exactly once in 1-factors Flb,o , l ∈ {0, . . . , k − 2}. Edges [m, (m − 1) mod (k − 1)], m ∈ {0, . . . , k − 1} are added to both near-1-factors of V0 and V1 in Flb,o , l ∈ {k − 1, . . . , 2k − 2}. Edges within V0 and V1 , respectively, are contained exactly once in Flb,o , l ∈ {k − 1, . . . , 2k − 2} by definition of near-1factors. ⊓ ⊔ Again, F0b,o is based on F0bip and, therefore, none of the differences i − j or j − i is equal to 1 if k > 1. Figure 4.4 illustrates the binary 1-factorization of K2k with k = 5. We can construct near-1-factorizations according to F b,e and F b,o of K4k−1 and K4k+1 by simply adding a dummy node, constructing the corresponding 1-factorization of K4k and K4k+2 , and considering each node matched with the dummy node as unmatched. Again, no difference in the first near-1-factor is equal to 1 if and only if k ∈ N, k > 1. 4.1.3 Ordered Symmetric 2-Factorization of 2K2k+1 A 2-factor of graph G = (V, E) is a set of edges E ′ ⊆ E such that each node i ∈ V is incident to exactly two edges e, e′ ∈ E ′ , e = e′ . A 2-factorization of G is a partition of its edges into 2-factors (see Franek and Rosa [39] for details). An ordered 2-factorization is a 2-factorization having its 2-factors ordered. The complete multi-graph 2Kn is a graph on |V | = n nodes having exactly two edges incident with each pair of nodes. Kn , n odd, is known to have a 2-factorization as outlined in Burling and Heinrich [15]. Hence, 2Kn , n odd, has one, as well. An oriented 2-factorization is a 2-factorization where each edge e ∈ E is given an orientation. Definition 4.3. A symmetric 2-factorization of 2Kk , k odd, is an oriented 2-factorization where edges corresponding to the same pair of nodes are given opposite orientations. We construct a symmetric 2-factorization 2F of 2Kk , k odd, as follows:

4.1 Factorizations

65

0

5

0

5

0

5

0

5

1

6

1

6

1

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1

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2

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3

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9

F0o,b 2

F1o,b

3

7

8

F2o,b

2

3

7

6

2

8

3

7

6

4

1

F3o,b

5

0

6

4

9 1

4

9 1 5

0

7

3

2

F6o,b 2

8

7

3

6 4

1

9 5

0 F7o,b

9 5

0

F5o,b

F4o,b

8

4

1

8

6 9 5

0 F8o,b

Fig. 4.4. Binary 1-Factorization of K10

2F = {2F0 , . . . , 2Fk−2 } , where 2Fl = {[m, (m + 1 + l)mod k]0 } k−1 − 1 , m ∈ {0, . . . , k − 1} , ∀ l ∈ 0, . . . , 2 2Fl+ k−1 = {[(m + 1 + l)mod k, m]1 } 2 k−1 − 1 , m ∈ {0, . . . , k − 1} . ∀ l ∈ 0, . . . , 2

66


Here, [i, j]k , i, j ∈ V , k ∈ {0, 1}, identifies the edge between nodes i and j and having index k being oriented i → j. Theorem 4.2. 2F is a symmetric 2-factorization of 2Kk , k odd. Proof. We show that 2F0 , . . . , 2F k−1 −1 forms a 2-factorization of 2

G′ := (V, {[i, j]0 | i, j ∈ V, i < j}). Obviously, each node has degree equal to two in 2-factor 2Fl unless k −1 2 holds which is impossible since k is odd and l is integer. For each pair i, j ∈ V , i < j, either [i, j]0 is contained in 2Fj−i−1 if is contained in 2Fk−1−(j−i) if j − i > k−1 j − i ≤ k−1 2 or [j, i] 2 . 0 Consequently, 2F k−1 , . . . , 2Fk−2 forms a 2-factorization of G′ := 2 (V, {[i, j]1 | i, j ∈ V, i < j}). Obviously, both edges incident to a pair i, j ∈ V have opposite orientations by definition. ⊓ ⊔ (m + 1 + l)mod k = (m − l − 1)mod k ⇔ l =

Note that each pair i, j ∈ V being matched in 2F0 has difference |i − j| = 1 as can be observed in figure 4.5. Furthermore, note that each node i is incident to exactly one ingoing edge [j, i]k , j ∈ V, k ∈ {0, 1}, and to exactly one outgoing edge [i, j]k , j ∈ V, k ∈ {0, 1}, in each 2factor 2Fl , l ∈ {0, . . . , k − 2}. Hence, each 2-factor consists of oriented circles.

2

3

1

2

4 0

3

1

4 0

2F0

2

3

1

2

4

1

0 2F1

3

4 0

2F2

Fig. 4.5. Symmetric 2-factorization of 2K5

2F3

4.2 Group-Balanced Tournaments

67

4.2 Group–Balanced Single Round Robin Tournaments In this section we provide several characteristics of group-balanced single RRTs. Additionally, we give a necessary and sufficient condition for n and |S| such that a corresponding group-balanced single RRT exists. Remark 4.1. The set of group-balanced single RRTs having n teams and |S| strength groups is a subset of the set of group-changing single RRTs having n teams and |S| strength groups. Hence, given n and |S| such that there is no corresponding group-changing single RRT then there neither is a corresponding group-balanced single RRT. Theorem 4.3. In a group-balanced single RRT the difference of two periods pj and pj where team i plays against teams j and j with S(j) = n S(j), respectively, is |pj − pj | = k|S| with k ∈ 0, . . . , |S| −1 .

Proof. Suppose team i plays against teams j and j with S(j) = S(j) in periods pj and pj with pj < pj and pj − pj = k|S| + l with k ∈ n − 1 and l ∈ {1, . . . , |S| − 1}. Then one of the following two 0, . . . , |S| cases holds. I

p −pj

There are less than j|S| − 1 matches of team i against teams of SS(j) in periods pj + 1, . . . , pj − 1. Then, there is at least one pair (p, p) with p, p ∈ pj , . . . , pj , p < p such that team i plays against

teams in S(j) in p and p, team i does not play against any team in

S(j) in any period p′ with p′ ∈ p + 1, . . . , p − 1 , and p − p > |S|. Hence, team i plays more than once against teams of at least one strength group Sk , k = S(j), in periods p + 1, . . . , p + |S|. p −pj

II There are more than j|S| − 1 matches of team i against teams of SS(j) in periods pj + 1, . . . , pj − 1. Then, there is at least one pair (p, p) with p, p ∈ pj , . . . , pj , p < p such that team i plays against teams in S(j) in p and p and p − p < |S|.

In both cases the single RRT is not group-balanced.

⊓ ⊔

Theorem 4.4. In a group-balanced single RRT each match of team i against S(i) = S(j) is carried out in period p = k|S| − 1, team j with n k ∈ 1, . . . , |S| − 1 .

68


Proof. According to theorem 4.3 the first period p containing a match between team i and an other team of strength group SS(i) determines the set of periods containing all matches between team i and teams of strength group SS(i) . If p = |S| − 1 then one of the following two cases holds. If p ∈ {|S|, . . . , n − 2} then team i plays twice against a team of at least one strength group Sk , k = S(i), in periods 0, . . . , |S| − 1. II If p ∈ {0, . . . , |S| − 2} then team i plays twice against a team of at least one strength group Sk , k = S(i), in periods n−|S|−1, . . . , n−2. I

⊓ ⊔

In both cases the single RRT is not group-balanced. Theorem 4.5. There is no group-balanced single RRT where

n |S|

is odd.

Proof. theorem 4.4 in each period p with p = k|S| − 1, According to n k ∈ 1, . . . , |S| − 1 , only matches between teams i and j with S(i) = n S(j) are carried out. If the number of teams |S| in a strength group n Sk is odd then no more than |S| − 1 teams of Sk can play in those periods. ⊓ ⊔ Theorem 4.6. In a group-balanced single RRT S(oi,p ) = S(oj,p ) holds for each period p and for each pair of teams (i, j) with S(i) = S(j). Proof. Assume there are teams i and j, S(i) = S(j), and a period p such that S(oi,p ) = S(oj,p ). Then, exactly one team of SS(oi,p ) plays against team j in period p′ with max{0, p−|S|+1} ≤ p′ ≤ p+max{0, |S|−p−1} and p′ = p, according to theorem 4.3. Obviously, |p′ − p| < |S| and, hence, |p′′ − p| = k|S|, k ∈ N, holds for each period p′′ where j plays against a team of SS(oi,p ) according to theorem 4.3. Then, team oi,p plays against i and j in two periods having distance not equal to k|S| for any k ∈ N which is infeasible since S(i) = S(j). ⊓ ⊔ Definition 4.4. A pairing of strength groups is a mapping σ : S → S such that σ(σ(Sk )) = Sk for each k ∈ {0, . . . , |S| − 1}. Theorem 4.7. There is no group-balanced single RRT where |S| is odd. Proof. According to theorem 4.6 there is a pairing of strength groups σp in each period p such that for two strength groups Sk , Sl , σp (Sk ) = Sl , each team in Sk plays against a team in Sl in p. Accordingto theorem n − 1 . Then, no 4.4 σp (Sk ) = Sk in period p = k|S| − 1, k ∈ 1, . . . , |S| σp exists if |S| is odd. ⊓ ⊔

4.2 Group-Balanced Tournaments

69

In order to construct a single RRT we have to arrange matches between each pair of teams and, therefore, pairings of strength groups such that each strength group is paired with each other strength group. This can be represented as 1-factorization of the complete graph K|S| where nodes correspond to strength groups and a 1-factor corresponds to a pairing. Two strength groups Sk , Sl , k = l, have to be paired exactly the amount of times needed to let each team of Sk play against each n team of Sl . This number is known to be |S| from the cardinality of n introduced n a 1-factorization of the complete bipartite graph K |S| , |S| in section 4.1.1. Furthermore, σp (Sl ) = Sl for each p = k|S| − 1, n k ∈ 1, . . . , |S| − 1 , l ∈ {0, . . . , |S| − 1} (since only matches between teams of identical strength groups can be carried out in these periods). Accordingly, the construction scheme proposed in the following has two stages. In the first stage a schedule is constructed which prescribes teams of a specific strength group Sk to play against teams of an other strength group Sl , l = k, or to play against teams of the same strength group Sk , respectively. The result, namely a strength group schedule, is exemplarily represented in table 4.1.

Table 4.1. Strength group schedule for n = 16, |S| = 4 k

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14

0 1 2 3

3 2 1 0

2 3 0 1

1 0 3 2

0 1 2 3

3 2 1 0

2 3 0 1

1 0 3 2

0 1 2 3

3 2 1 0

2 3 0 1

1 0 3 2

0 1 2 3

3 2 1 0

2 3 0 1

1 0 3 2

Line 2 to 5 correspond to strength groups S0 to S3 . For each strength group Sk , k ∈ {0, . . . , 3}, the strength group Sl , l ∈ {0, . . . , 3}, being paired with Sk in period p is given in the line corresponding to Sk and in the column corresponding to p. In the second stage we arrange matches between teams according to a given strength group schedule and to 1-factorizations introduced in section 4.1. Theorem 4.8. A group-balanced single RRT can be arranged if and n is even. only if |S| is even and |S| Proof. We show that a group-balanced single RRT can be arranged if n is even. Then, considering theorems 4.5 and 4.7 |S| is even and if |S| theorem 4.8 follows.

70


We first construct the pairing σp of strength groups for each period p. According to theorem 4.4 σp (Sl ) = Sl for each p = k|S| − 1, k ∈ n 1, . . . , |S| − 1 , l ∈ {0, . . . , |S| − 1}. Additionally, we arrange pairings of strength groups according to a 1-factorization of K|S| in periods 0 to |S| − 2. This is possible if and only if |S| is even. Naturally, the 1-factorization structure means each strength group being paired with each other strength group exactly once and each strength group being contained in each pairing σp , p ∈ {0, . . . , |S| − 2} exactly once. Next, we set σp+k|S| = σp , for each p ∈ n n − 1 . Hence, we obtain exactly |S| {0, . . . , |S| − 2} and k ∈ 1, . . . , |S| pairings containing a specific pair Sk , Sl , k = l, of strength groups. For each pair of strength groups Sk , Sl , k = l, we arrange all matches between teams of Sk and Sl in a way representable as a 1-factorization n as shown in section 4.1.1. n of K |S| , |S| Next, we arrange all group inherent matches in periods p with p = n k|S| − 1, k ∈ 1, . . . , |S| − 1 by arranging a 1-factorization for the n corresponding to each strength group S complete graph K |S| k where n nodes represent teams of Sk . This is possible if and only if |S| is even (see section 4.1.2 for details). The result is a group-balanced single RRT since: • Each strength group is contained in each σp , p ∈ {0, . . . , n − 2}. Due to the 1-factorization structure according to teams of one strength group and teams of two paired strength groups, respectively, each team plays exactly once per period. • Each pair of teams meets exactly once due to the 1-factorization structure according to teams of one strength groups and teams of two paired strength groups, respectively. • No team plays more than once against teams of the same strength group within |S| consecutive periods since identical pairings have n distance of k|S|, k ∈ 0, . . . , |S| − 1 periods by construction.

Hence, theorem 4.8 holds.

⊓ ⊔

4.3 Group–Changing Single Round Robin Tournaments In this section we discuss cases of n and |S| where no group-balanced single RRTs exists and give construction schemes for group-changing single RRT. We adopt the basic idea of pairings of strength groups

4.3 Group-Changing Tournaments

71

from section 4.2. However, we have to extend this concept in order to allow fairness according to changing strength groups for more cases than those given in section 4.2. n Remark 4.2. If |S| is even and |S| is even a group-changing single RRT can be arranged by construction proposed for group-balanced single RRTs.

Remark 4.3. If |S| = 2 a group-changing single RRT is group-balanced as well and, therefore, no group-changing single RRT with |S| = 2 exists if n2 is odd. Lemma 4.1. If |S| = 4k + 3, k ∈ N+ , a group-changing single RRT can be arranged. Proof. We construct a group-changing single RRT given n and |S| = 4k + 3, k ∈ N+ . First, we construct a binary near-1-factorization according to F b,e as introduced in section 4.1.2 on the complete graph K|S| . We interpret each near-1-factor as a pairing σ of strength groups (see definition 4.4). Additionally, we define σ(Sk ) = Sk if the node not matched in the near-1-factor corresponds to Sk . The pairing resulting b,e n −1 . from Fl is assigned to period p = k|S| − 1 − l, k ∈ 1, . . . , |S| Thus, periods 0 to n − |S| − 1 are assigned to pairings containing each n − 1 times. pair (Sk , Sl ), k, l ∈ {0, . . . , |S| − 1}, exactly |S| We arrange matches in periods 0 to n − |S| − 1 according to the pairing σp assigned to p, i.e. a match of i against j can not be carried out in period p if S(i) is not paired with S(j) in p. For each pair of strength groups Sk , Sl we arrange matches according to the 1-factorization of n n n as given in section 4.1.1. Since we can arrange only K |S| , |S| |S| − 1 out for each pair of strength groups. 1-factors we leave F bip n 2|S|

n −1 Furthermore, each strength group is paired with itself exactly |S| times. Hence, we can arrange all matches between teams of the same n (see section 4.1.2). strength group according to a 1-factorization of K |S| n n of K |S| corFinally, all matches contained in 1-factors F bip n , |S| 2|S|

responding to each pair of strength groups have to be arranged in periods n − |S| to n − 2. We construct a symmetric 2-factorization of 2K|S| according to section 4.1.3 and assign factor 2Fp to period n − |S| + p, p ∈ {0, . . . , |S| − 2}. If [i, j]k is contained in 2-factor 2Fp , n teams p ∈ {0, . . . , |S| − 2}, we arrange matches between the first 2|S|

72


n n of Si and the last 2|S| teams of Sj in period n − |S| + p. Note that |S| is even since |S| is odd while n is even. Now, we have a single RRT with changing opponents’ strengths:

• Each team plays exactly once per period. For each period p, p ∈ {0, . . . , n − |S| − 1}, this is obvious due to the 1-factor structure within pairs of strength groups. In each remaining period p, p ∈ {n − |S|, . . . , n − 2}, each team plays exactly once since each 2factor is composed of oriented circles (see section 4.1.3). Hence, for each strength group the outgoing arc covers the first half of teams while the ingoing arc covers the second half. • Each pair of teams meets exactly once. Obviously, no pair of teams plays twice in periods p, p ∈ {0, . . . , n − |S| − 1}, due to the 1factorization structure between each pair of strength groups and bewithin single strength groups, respectively. The 1-factors F bip n 2|S|

tween each pair of strength groups missing from periods 0 to n−|S|−1 are exactly covered by both arcs between a pair of strength groups in 2K|S| . • No team plays against teams of the same strength group in two consecutive periods. In each time window [p, p + |S| − 1], p ∈ {0, . . . , n − 2|S|}, each team plays exactly once against each strength group due to repeating sequence of pairings. In periods p, p ∈ {n − |S|, . . . , n − 2}, each strength group is paired twice with each other strength group. However, corresponding arcs have opposite orientation and, hence, the set of teams involved in both pairs are disjoint. Therefore, in periods p, p ∈ {n − |S|, . . . , n − 2} each team plays exactly once against each other strength group and, hence, there is no violation of changing opponents’ strengths in periods p and p + 1, p ∈ {0, . . . , n − |S| − 2} and p ∈ {n − |S|, . . . , n − 2}, respectively. Note that 1-factor F0b,e chosen for p = n−|S|−1 does not contain any pair (Si , S(i+1) mod |S| ) (see section 4.1.2) if |S| > 3. 2F0 chosen for period p = n − |S| exclusively contains pairs of this form. Therefore, no team can play against the same strength group in periods n − |S| − 1 and n − |S|. Hence, lemma 4.1 holds.

⊓ ⊔

Lemma 4.2. If |S| = 4k + 1, k ∈ N+ a group-changing single RRT can be arranged. Proof. The proof is analogous to the proof of lemma 4.1. The only difference is employing the binary 1-factorization of K2k , k odd, F b,o

4.3 Group-Changing Tournaments

73

given in section 4.1.2 instead of the binary 1-factorization F b,e in order to establish pairings for periods 0 to n−|S|−1. Each conclusion follows as above. ⊓ ⊔ Theorem 4.9. According to lemmas 4.1 and 4.2 a group-changing single RRT can be arranged if |S| > 3 and odd. Theorem 4.10. If can be arranged.

n |S|

is odd and |S| > 2 a group-changing single RRT

n Proof. Given n and |S| > 2 with |S| odd we construct a group-changing single RRT. First, we construct an ordered 1-factorization of K|S| and associate 1-factors with pairings such that we obtain pairing σ |S| −1 2 − 1 . having Sk paired with Sk+1 for each k ∈ 2l | l ∈ 0, . . . , |S| 2 n −2 , Then we assign σp to periods p + k|S| + 1 for k ∈ 0, . . . , |S| p ∈ {0, . . . , |S| − 2}. ! n Additionally, we assign σp to period p + |S| − 1 |S| + 1 for each ! n ′ + ′ p ∈ 0, . . . , |S| to period p − 2 and we assign σ − 1 |S| p 2 |S| for each p′ ∈ |S| 2 , . . . , |S| − 2 . Hence, each pair of strength groups n except those contained in σ |S| −1 is arranged exactly |S| times. We can 2

construct 1-factorization F bip according to section 4.1.1 for the teams contained in each of those pairs of strength groups. Pairing σ |S| −1 is contained exactly

n |S|

2

− 1 times. Therefore, we can arrange all 1-factors

bip of F bip but F |S| . Consequently, all matches between pairs of strength 2

−1

n + k, groups are arranged except between teams i and j, i = S(i) |S| n n n j = (S(i) + 1) |S| + (k − 1) mod |S| , S(i) even, k ∈ 0, . . . , |S| − 1 . n − 1 , we To the remaining periods p, p = k|S|, k ∈ 0, . . . , |S| assign the pairing having each strength group paired with itself. Since n n |S| is odd we can construct a near-1-factorization with |S| near-1-factors for the set of teams in each strength group according to section 4.1.2. Naturally, in each strength group each team is not contained in a near1-factor exactly once. Team i ∈ Sk , k even, in the near not contained n 1-factor assigned to period p = l|S|, l ∈ 0, . . . , |S| − 1 is arranged to play against team j ∈ Sk+1 not contained in the near-1-factor assigned to period p. Formally, we arrange 1-factors Flb,o , l ∈ {k − 1, . . . , 2k − 2}, according to section 4.1.2 for each pair (Sm , Sm+1 ), m even. Here, Sm

74


and Sm+1 correspond to V0 := {0, . . . , k − 1} and V1 := {k, . . . , 2k − 1} of K2k , k odd, respectively. This results in a group-changing single RRT: • Each team plays exactly once per period. For period p, p = k|S|, k ∈ n 0, . . . , |S| − 1 , this is obvious due to the 1-factor structure within n pairs of strength groups. In periods p, p = k|S|, k ∈ 0, . . . , |S| −1 , each team but one per strength group plays exactly once due to the near-1-factor structure within each strength group. Team i, S(i) even, not playing against an other team of SS(i) in period p, p = k|S|, n − 1 , plays against the team j ∈ SS(i)+1 not playing k ∈ 0, . . . , |S| n odd) against an other team of SS(i)+1 . Since |S| is even (n even, |S| each of these teams plays exactly once. • Each pair of teams meets exactly once. This is obvious for matches between teams i and j, S(i) < S(j), (S(i) odd ∨ S(i) + 1 = S(j)), of different strength groups due to the 1-factorization structure between each pair of strength groups. Furthermore, matches between teams i and j, S(i) = S(j), are carried out exactly once due to the near-1-factorization structure within strength groups. Matches of teams i and j (S(i) even ∧S(i)+ 1 = S(j))are composed of Fkbip , |S| 2

k =

− 1, in periods p, p ∈

|S| 2

n + l|S| | l ∈ 0, . . . , |S| −2

bip F |S|

, and

arranged between pairs of unmatched nodes in near-1-factors n −1 . in periods p, p ∈ l|S| | l ∈ 0, . . . , |S| • No team plays against teams of the same strength group in two consecutive periods. This is obvious for pairs Sk , Sl , k < l, (k odd ∨ k + 1 = l) of strength groups since those are arranged in periods having distance no less than |S| − 1 by construction. |S| − 1 > 1 for |S| > 2. Matches within pairs Sk , Sl , k < l (k even ∧ k + 1 = l), of strength groups are arranged in periods p ∈ P in with n |S| in + l|S| | l ∈ 0, . . . , −2 ∪ P := 2 |S| n l|S| | l ∈ 0, . . . , −1 |S| |S| 2n = l | l ∈ 0, . . . , −2 . 2 |S| 2

−1

Obviously, pairwise distance is no less than 2 if |S| > 2.

4.4 Complexity

Hence, theorem 4.10 holds.

75

⊓ ⊔

Remark 4.4. By enumeration: If n = 6 and |S| = 3 no group-changing single RRT exists but if n ∈ {12, 18} and |S| = 3 a group-changing single RRT can be arranged. We conjecture that a group-changing single RRT exists for |S| = 3 and each n = 6(k + 1), k ∈ N+ .

4.4 Complexity As outlined in section 2.1 there are several applications for associating cost ci,j,p with each match of team i at home against team j in period p. For the sake of convenience we repeat the definition of the single RRT problem given in section 2.1. Definition 2.3. Given a set T , |T | = n, and a set P , |P | = n − 1, each triple (i, j, p) ∈ T × T × P, i = j, represents a match of team i against team j at i’s home in period p. Costs ci,j,p are given for each match as well. A feasible solution to the single RRT problem corresponds to a set of n(n−1) triples such that (i) for each pair (i, j) ∈ T × T, i < j, 2 exactly one triple of form (i, j, p) or (j, i, p) with p ∈ P is chosen and such that (ii) for each pair (i, p) ∈ T × P exactly one triple of form (i, j, p) or (j, i, p) with j ∈ T \{i} is chosen. The problem is to find a feasible solution having the minimum sum of chosen triples’ cost. The single RRT problem has been proven to be NP-hard independently in Briskorn et al. [12] and in Easton [31]. We introduce two minimum cost problems corresponding to strength group requirements as introduced in section 3.2.2 and definitions 4.1 and 4.2. Definition 4.5. Given a set T , |T | = n even, of teams, a set of periods P , |P | = |T | − 1, a number |S| of strength groups and cost ci,j,p associated with each match of team i ∈ T at home against team j ∈ T , j = i, in period p ∈ P the group-balanced single RRT problem is to find the group-balanced single RRT having the minimum sum of arranged matches cost. Definition 4.6. Given a set T , |T | = n even, of teams, a set of periods P , |P | = |T | − 1, a number |S| of strength groups and cost ci,j,p associated with each match of team i ∈ T at home against team j ∈ T , j = i in period p ∈ P the group-changing single RRT problem is to find

76


the group-changing single RRT having the minimum sum of arranged matches cost. Theorem 4.11. The group-balanced single RRT problem is NP-hard even if |S| is fixed. We proof theorem 4.11 by reduction from single RRT problem. Proof. Given a single RRT problem by a set of teams T ′ , |T ′ | = n′ , a set of periods P ′ , |P ′ | = n′ − 1, and cost c′i′ ,j ′ ,p′ , i′ , j ′ ∈ T ′ , i′ = j ′ , p′ ∈ P ′ , we construct a group-balanced single RRT problem with n teams and |S| strength groups as follows. Let n = n′ |S|. We follow the idea of pairings of strength groups in each period given in section 4.2. We set cost

ci,j,p =

  M   

if

S(i) = S(j) = 0, p = k|S| − 1,

n k ∈ 1, . . . , |S| −1 ,

 S(i) = S(j) = 0, p = (p′ + 1)|S| − 1, p′ ∈ P ′ , c′i,j,p′ if     0 else, with M = i′ ∈T ′ j ′ ∈T ′ ,j ′ =i′ p′ ∈P ′ ci′ ,j ′ ,p′ . Obviously, a group-balanced single RRT having cost less than M is provided by the construction scheme given in the proof of theorem 4.8. Each solution having cost less than M provides s of a single RRT n teams of S0 in periods p with p = k|S| − 1, k ∈ 1, . . . , |S| − 1 . Next, it can be easily seen that s is optimal for the original minimum cost single RRT problem by contradiction. If there is a single RRT s′ having less cost than s according to the single RRT problem we can exchange s by s′ in the group-balanced single RRT. Trivially, this leads to a group-balanced single RRT having less cost. ⊓ ⊔ Lemma 4.3. The group-changing single RRT problem is NP-hard even if |S| > 3 is odd and fixed. Again, we give a reduction from the single RRT problem. The idea is quite the same as for theorem 4.11. Hence, a sketch of the proof suffices. Proof. Given a single RRT problem we construct a group-changing single RRT problem with n teams and |S| > 3, |S| odd, strength groups as follows. Let n = n′ |S|. We follow the idea of pairings of strength groups by near-1-factorizations according to F e,b and F o,b (depending

4.5 Summary

77

on the number of strength groups |S|). Note that S0 is matched with n itself in period |S| 2 − 1 + k|S| for each k ∈ 0, . . . , |S| − 2. We set cost  ′  M if S(i) = S(j) = 0, p = |S|  2 − 1 + p |S|,   n   p′ ∈ 0, . . . , |S| −2 ,   ′ ci,j,p = c′i,j,p′ if S(i) = S(j) = 0, p = |S| 2 − 1 + p |S|,    n  −2 , p′ ∈ 0, . . . , |S|     0 else, with M = i′ ∈T ′ j ′ ∈T ′ ,j ′ =i′ p′ ∈P ′ ci′ ,j ′ ,p′ . Obviously, a group-changing single RRT having cost less than M is provided by the construction scheme given in the proof of lemmas 4.1 and 4.2, respectively. Each solution having cost less than M provides |S| ′ a single RRT s ofteams of S0 in periods p = 2 − 1 + p |S| with

n −2 . p′ ∈ 0, . . . , |S| Obviously, s is optimal for the original single RRT problem.

⊓ ⊔

Lemma 4.4. The group-changing single RRT problem is NP-hard even if |S| is even and fixed. Again, we give a reduction from the single RRT problem. The idea is quite the same as for theorem 4.11. Proof. Since the number of teams n′ of the single RRT problem is restricted to even values and |S| is given even we can reduce the single RRT problem exactly as in the proof of theorem 4.11. ⊓ ⊔ Theorem 4.12. According to lemmas 4.3 and 4.4 the group-changing single RRT problem is NP-hard even if |S| = 3 is fixed.

4.5 Summary In this section we pick up a common idea to achieve fairness among teams competing in a single RRT. Although strength groups have already been proposed in several works there is no answer to the question for which values of n and |S| fair schedules can be constructed. We investigate two degrees of fairness: group-changing single RRTs and group-balanced single RRTs. We proof a necessary and sufficient condition for n and |S| to allow a group-balanced single RRT. Furthermore, we show how to decide for

78


almost all cases whether a group-changing single RRT is possible or not. The remaining cases are n = 6k, k ∈ N, and |S| = 3 and we strongly conjecture a group-changing RRT to be possible if and only if k > 1.

5 Home-Away-Pattern Based Branching Schemes

5.1 Motivation This chapter focuses on branching schemes in line with decomposition schemes following the first-break-then-schedule idea (see section 2.4.2 for details). We consider the LP relaxation of the problem at hand, solve it to optimality, and branch on the venue of team i in period p. Accordingly, we construct a branching tree where a path from the root node to a leaf either can be pruned or represents a full HAP set. Branching on a subset of variables is a well known generalization of {0, 1}-branching on binary variables. Here, we choose a subset Qi,p of that the venue of team i is fixed in p by forcing variables such x = 1 or x∈Qi,p x∈Qi,p x = 0, respectively. Hence, Qi,p := {xi,j,p | j ∈ T \ {i}}. Deciding the venue of a specific team and period is part of almost all variants of RRTs presented in chapter 2. Furthermore, solutions to LP relaxations to most problems presented in chapters 2 and 3 do not imply a consistent decision about venues (which means there probably is a i ∈ T and a p ∈ P such that x∈Qi,p x ∈ {0, 1}). Therefore, the branching idea is applicable as well as reasonable to almost all RRT problems. Let Qi,p := {xj,i,p | j ∈ T \ {i}}. Then, Qi,p + Qi,p = 1 due to constraint (2.8) and, hence, Qi,p = 1 ⇔ Qi,p = 0. Consequently, we implement each branching step by fixing subsets of variables Qi,p and Qi,p , respectively, to 0. In section 5.2 we treat the general case where no restriction concerning venues is given. Section 5.3 considers the case where we require the minimum number of breaks. In both cases defining a HAP set by branching can not guarantee binary solutions to the problem’s LP relaxation. Table 5.1 represents a fractional solution on the left hand

80

5 HAP Based Branching Scheme

side. Each match in periods 1 and 2 has variable value 0.5, for example. Matches in period 3 have variable value 1. The unique corresponding HAP set (having the minimum number of breaks) is shown on the right hand side. However, we refuse to develop a full branching scheme here in order to emphasize the power of the branching idea as a guiding scheme for the first levels of a branching tree. Consequently, if a node providing a full HAP set can not be pruned we solve the corresponding IP problem using the standard solver Cplex. Table 5.1. Fractional Solution Without Candidate for HAP set branching Period MDs

1

2

3

Team 1 2 3

1-2 1-4 2-1 2-3 2-4 3-4 3-2 4-3 4-1 1-3

Variable 0.5 0.5 0.5 0.5 1

1 2 3 4

0 1 0 1

10 00 11 01

Consider an IP problem corresponding to a HAP set. In section 2.4.2 we conjecture the corresponding LP relaxation not to be able to provide a larger number of matches than the IP problem does. If this conjecture holds the LP relaxation corresponding to an infeasible HAP set is proofed infeasible and, consequently, the node is pruned before the IP problem is solved. Then, no IP problem corresponding to an infeasible HAP set has to be solved.

5.2 General Home-Away-Pattern Sets First, we like to emphasize the importance to recognize infeasible HAP sets in order to keep computational effort low. Clearly, infeasible HAP sets correspond to infeasible nodes and subtrees, respectively, of the branching tree. Several necessary conditions for HAP sets to be feasible are considered by our branching scheme. This is lined out in section 5.2.1. In the following we specify several approaches by proposing alternative strategies to choose the next branching candidate as well as node order strategies in sections 5.2.2 and 5.2.3.

5.2 General Home-Away-Pattern Sets

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5.2.1 Achieving Feasible Home-Away-Pattern Sets As lined out in section 2.4.2 HAP sets can not easily proofed either feasible or infeasible. There is no simple characterization of feasible HAP sets. Therefore, avoiding infeasible HAP sets is the challenging part here. We show that our branching scheme constructs HAP sets fulfilling two necessary conditions hereafter. We define the partial HAP as a generalization of the HAP. Definition 5.1. A partial HAP of team i is a string containing 0 at slot p if i plays at home in period p, containing 1 at slot p if i plays away in p, and containing ∗ at slot p if the venue of i in p is not decided. Consequently, a partial HAP set is a set of n partial HAPs being assigned to teams. Each node in our branching tree represents a partial HAP set. The partial HAP set corresponding to the root node exclusively contains asterisks. Following a path from the root node to a leaf we replace a ∗ either by 0 or by 1 in each step. A partial HAP set is called feasible in the remainder if we can obtain a feasible HAP set by replacing all asterisks. We adopt two necessary conditions for HAP sets to be feasible given in section 2.4.2 in order to consider partial HAP sets. (i) Partial HAPs of two teams must be different or contain at least one ∗. (ii)Each column of a partial HAP set must not contain more than n2 zeros and must not contain more than n2 ones. Theorem 5.1. When we branch on candidates Qi,p ∈ {0, 1} we can not create a partial HAP set violating condition (i) or condition (ii). Proof. Proof is done by contradiction: (i) Suppose we construct two identical HAPs corresponding to teams i and j having no ∗. Then, those two HAPs have differed in exactly one slot p in the current node’s father. Hence, in each feasible solution according to the current node’s father these two teams play against each other in period p. Therefore, neither Qi,p nor Qj,p has been a branching candidate. (ii)Suppose (w.l.o.g.) that we construct column p having n2 + 1 zeros. Then, this column has exactly n2 zeros in the current node’s father. Therefore, each team having no zero must play away in each feasible solution. Therefore, there was no branching candidate Qi,p , i ∈ T . Hence, both conditions can not be violated.

⊓ ⊔

82


Moreover, we adapt a necessary condition for HAP sets to be feasible developed in Miyashiro et al. [64] to partial HAP sets. Let c0 (T ′ , p), c1 (T ′ , p), and c∗ (T ′ , p) be the number of zeros, ones, and asterisks, respectively, in column p in lines corresponding to teams in T ′ ⊆ T . Miyashiro et al. [64] propose the following condition for (non-partial) HAP sets.

p∈P

′

|T | min c0 (T , p), c1 (T , p) − ≥ 0∀T′ ⊆ T 2

′

′

(2.25)

The first term sums up the number of matches between teams in T ′ being possible according to the HAPs (see Miyashiro et al. [64] for ′ details). At least |T2 | matches must be possible in order to construct a single RRT. In order to consider partial HAP sets we propose the following modified condition.

p∈P

min

"

#

|T ′ | ′ ′ ′ , min c0 (T , p), c1 (T , p) + c∗ (T , p) 2 ′ |T | − ≥ 0 ∀ T ′ ⊆ T (5.1) 2

Note that (5.1) is identical to (2.25) if c∗ (T ′ , p) = 0 for each p ∈ P . If c∗ (T ′ , p) > 0 we can choose undecided venues of teams such that we maximize the number of possible matches between teams of T ′ in p. Therefore, if w.l.o.g. c0 (T ′ , p) < c1 (T ′ , p) we consider min {|c0 (T ′ , p) − c1 (T ′ , p)|, c∗ (T ′ , p)} teams having undecided venues to play at home in p, first. If c∗ (T ′ , p) ≤ |c0 (T ′ , p) − c1 (T ′ , p)| there are no teams having undecided venues left and the number of possible matches between teams ′ of T ′ in p is restricted to min {c0 (T ′ , p), c1 (T ′ , p)} + c∗ (T ′ , p) ≤ |T2 | in p. If c∗ (T ′ , p) > |c0 (T ′ , p)−c1 (T ′ , p)| following the idea described above we obtain identical numbers of teams playing at home and away, respectively. The remaining teams having undecided venues are chosen to play at home or away such that the overall number of teams playing ′ at home and away, respectively, differ by no more than 1. Then, |T2 |

matches between teams of T ′ are possible in p. Formally, we obtain this effect by the outer minimization in (5.1).


83

Theorem 5.2. When we branch on candidates Qi,p ∈ {0, 1} we can not create a partial HAP set violating (5.1). Proof. First, we show that the left hand side of (5.1) can be lowered by no more than 1 if we replace a single asterisk. Let (w.l.o.g.) team i’s entry in slot p be changed from ∗ to 0. • If c0 (T ′ , p) < c1 (T ′ , p) with T ′ ⊆ T , i ∈ T ′ , before the replacement the left hand side of (5.1) does not change since the inner minimization term is increased by 1 and c∗ (T ′ , p) is decreased by 1. • If c0 (T ′ , p) ≥ c1 (T ′ , p) and c∗ (T ′ , p) > c0 (T ′ , p) − c1 (T ′ , p) before the replacement the left hand side of (5.1) does not change due to the outer minimization. • If c0 (T ′ , p) ≥ c1 (T ′ , p) and c∗ (T ′ , p) ≤ c0 (T ′ , p) − c1 (T ′ , p) before the replacement the left hand side of (5.1) decreases by 1 since the inner minimization does not change and c∗ (T ′ , p) is decreased by 1. Consequently, the outer ′ minimization is decreased by one since ′ ′ c1 (T , p) + c∗ (T , p) ≤ |T2 | . Now, suppose we create a partial HAP set having |T ′ | |T ′ | ′ ′ ′ = − 1 p∈P min 2 2 , min (c0 (T , p), c1 (T , p)) + c∗ (T , p)

with T ′ ⊆ T by changing the entry for team i ∈ T ′ in period p from ∗ to 0 (in accordance with the reasoning above). Then, according to the partial father node’s ′ HAP set corresponding to the current |T | ′ , p), c (T ′ , p)) + c (T ′ , p) = |T ′ | . Further, min (c (T min 0 1 ∗ p∈P 2 2 more, the third case described above is given since it is the only one resulting in a decreasing left hand side of (5.1). Consequently, in each feasible solution each team having ∗ or 1 in period p plays away. Therefore, Qi,p is no branching candidate according to the father node’s optimal solution. ⊓ ⊔ Note that condition (i) is a special case of (5.1) with |T ′ | = 2. We conclude that two necessary conditions (ii) and (5.1) for partial HAP sets to be feasible can not be violated following our idea of branching candidates Qi,p ∈ {0, 1}. 5.2.2 Choice of Branching Candidates Given an optimal solution to the LP relaxation of a RRT problem introduced in chapters 2 and 3 each tuple (i, p) ∈ T × P with j∈T \{i} xi,j,p ∈ {0, 1} is candidate for the next branching step.

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Random Choice: Starting with a randomly chosen tuple (i, p) we sequentially search all Qi,p for fractional values and choose the first branching candidate being found. Hence, we interpret neither the LP relaxation’s optimal solution nor the cost structure. The main advantage of this strategy is its low run time consumption. Least Fractional: We define infi,p := min x, x x∈Qi,p x∈Qi,p as a measure of infeasibility of (i, p) in the current LP problem’s optimal solution. Among all branching candidates (i, p) we choose the one having lowest infi,p which means Qi,p is closest to binary. The idea is that forcing candidates having small infeasibility to feasibility causes only slight modifications to the LP relaxation’s optimal solution. Most Fractional: Among all branching candidates (i, p) we choose the one having highest infi,p which means Qi,p is closest to 0.5. As outlined in Achterberg et al. [1] the idea is to force feasibility for those tuples first where the LP solution implicates least tendency whether Qi,p to round to zero or to one. P Pseudo-Cost: We compute pseudo-cost chi,p = P

j∈T \{i} cj,i,p n−1

j∈T \{i} ci,j,p

n−1

and

representing average matches’ cost at home and cai,p = away, respectively, according to team i and period p. Among all branching candidates (i, p) we choose the one having lowest min{chi,p , cai,p }. This strategy aims at fixing those candidates first which propose low cost matches. We calculate chi,p and cai,p a single time before the branchand-bound (B&B) procedure starts and, hence, computational effort for considering pseudo-cost is identical to the one for least fractional and most fractional, respectively. Regret: We define regret ri,p as the cost of not choosing the venue having lower pseudo-cost for candidate (i, p): ri,p = |cai,p − chi,p |. Again, ri,p can be calculated beforehand and, therefore, does not increase computational effort. Pseudo-Cost Revisited: If we choose candidate (i, p) and fix the corresponding venue the set of possible matches of team j = i in period p is affected. A match of team i against team j at the venue ruled out for i can not be carried out in period p. Therefore, it is reasonable to adjust chi,p and cai,p according to the fixed venues on the path from the root node to the current node k. Adjusted pseudo-costs are denoted by − → k,a ck,h i,p and ci,p below. Let Vk denote the path from the root node to the current node. If team i is fixed to play at home or away in period p − → − → according to any branching step in Vk we denote this by Qi,p ∈ Vk or − → Qi,p ∈ Vk , respectively. Formally, modification of pseudo costs can then be stated as follows.


ck,h i,p ck,a i,p

85

→ ci,j,p (n − 1)chi,p − Q ∈− j,p Vk $ = − →$$ $ n − 1 − $ Qj,p | Qj,p ∈ Vk $ → ci,j,p (n − 1)cai,p − Q ∈− j,p Vk $ = − →$$ $ n − 1 − $ Qj,p | Qj,p ∈ Vk $

Obviously, we can redefine regret depending on the current node k by k,a k,a k,h k employing ck,h i,p and ci,p : ri,p = |ci,p − ci,p |. Then, rules “Pseudo-Cost” and “Regret” can be applied to modified pseudo-costs. 5.2.3 Node Order Strategy When working off the set of nodes of the branching tree we have to decide which node to explore next. We implement two well known node order strategies: • Depth First Search: The node which has been created last is explored first. This strategy minimizes the memory requirements. • Breadth First Search: We define a fitness for each node. The node having best fitness is explored first. We employ the lower bound value obtained from the father’s LP problem as fitness. This strategy requires more memory than depth first search does but mostly leads to shorter run times. For both of these strategies the question arises in which order nodes having the same father are explored. Here, the decision is made depending on the choice of branching candidates according to section 5.2.2. Random Choice: Since candidate Qi,p has been chosen randomly we choose the order of corresponding child nodes randomly as well. Least Fractional, Most Fractional: Given a chosen branching candidate (i, p) we explore the node corresponding to i playing at home in p first if x∈Qi,p x ≥ 0.5. Otherwise we explore the node corresponding to i playing away in p first. Pseudo-Cost (revisited), Regret: Given a chosen branching candidate (i, p) one of both child nodes corresponds to the venue of i in p having the lower (revisited) pseudo-cost. This node is explored first.

86


5.3 Minimum Number of Breaks Again, the basic idea is to branch on Qi,p . In opposite to general HAP sets as considered in section 5.2 the venues of team i in two periods are not independent here. Although there is no direct dependency of venues in consecutive periods the occurrence of identical venues in consecutive periods (namely a break) is restricted to an overall number of n − 2 for all teams, see sections 2.4 and 3.2.1. If team i has no break we say it has a break in the first period which is justified by the fact that the first and the last entry of the corresponding HAP are identical, then. In a RRT having the minimum number of breaks each team has exactly one break (see Miyashiro et al. [64]) and, therefore, we can specify each team’s HAP by venue and period of its unique break. Consequently, we branch on venue and period of a specific team’s break, here. Branching candidates are teams whose break is not fully specified by the optimal solution to the current node’s LP problem. We implement a specific break by fixing match variables to 0 according to the break’s venue and period. If we branch on team i to have a home-break in period p then we can fix to zero half the match variables corresponding to i as follows. Qi,p′ = 0 ∀p′ ∈ P, ((p′ < p ∧ p − p′ even) ∨ (p′ > p ∧ p′ − p odd)) Qi,p′ = 0 ∀p′ ∈ P, ((p′ < p ∧ p − p′ odd) ∨ (p′ ≥ p ∧ p′ − p even)) Consequently, fixing variables according to an away-break for team i in period p is done the other way round. Qi,p′ = 0 ∀p′ ∈ P, ((p′ < p ∧ p − p′ even) ∨ (p′ > p ∧ p′ − p odd)) Qi,p′ = 0 ∀p′ ∈ P, ((p′ < p ∧ p − p′ odd) ∨ (p′ ≥ p ∧ p′ − p even)) While we can represent team i having a break in p by fixing match variables (as seen above) we can not represent i not having a break in p by fixing match variables. Therefore, we propose a branching strategy where each subproblem is represented by a fixed break. Consequently, given a chosen branching candidate i ∈ T we create a child node for each single break (defined by venue and period) which can be assigned to i. Considering that each team has exactly one break in a feasible solution we obtain a partition of solution space corresponding to the current node into solution spaces corresponding to its child nodes. Obviously, as lined out in Briskorn and Drexl [11] this means a number of up to 2(n − 1) child nodes.

5.3 Minimum Number of Breaks

87

5.3.1 Achieving Feasible Home-Away-Pattern Sets Again, it is of great importance to recognize infeasible HAP sets and, moreover, avoid the construction of corresponding nodes in order to save run time. Several necessary conditions for HAP sets to be feasible are considered in section 5.2.1. Below we propose several strategies to calculate the set of possible breaks (each defined by period and venue) for a given branching candidate i ∈ T . Since we construct a child node for each break being considered possible the challenging part is to keep this set as small as possible. On the other hand, neglecting possible breaks results into inadmissible reduction of solution space and, therefore, must be avoided. No Restrictions: Here, we simply create a child node for each period and venue without consideration of breaks already fixed for other teams. Therefore, each node has 2(n − 1) children if it is not pruned. No Break Twice: The set of possible breaks can be reduced by taking into account the breaks already assigned to teams on the path from the root node to the current node. Two teams having identical breaks leads to both teams having identical HAPs. As lined out in section 2.4.2 an infeasible HAP set follows. Therefore, no break is assigned to more than one team. Consequently, the set of possible breaks can be reduced by all breaks being already assigned to a team on the path from the root node to the current node. Break Sequences: It is known from, e.g., Miyashiro et al. [64] that there are either no or two breaks in each period. Hence, we have to choose n2 − 1 periods (additional to the first period) where breaks occur and assign teams to both breaks corresponding to one of these periods in order to construct a HAP set. We refer to these periods as break periods in the remainder. Given a HAP set a specific assignment of each HAP to a team does not influence the HAP set’s feasibility. The HAP set is fully specified by a set of n2 break periods as far as feasibility is concerned. The set of break periods in ascending order is referred to as break sequence in the remainder. In de Werra [19] break sequences corresponding to the special class of canonical 1-factorizations are studied. − → Again, let k and Vk denote the current node of the branching tree and the path from the root node to the current node. Additionally, let − → br → and P− nbp,− → be the number of breaks fixed in period p on Vk and the Vk Vk − → set of periods where at least one break has been fixed in on Vk . Then, we can apply the following rules I to III as outlined in Briskorn and Drexl [11] in order to decide which breaks must be considered possible.

88


I

A home-break (away-break) in the first period is possible if and only if no home-break (away-break) has been set in the first period on − → Vk . → = 1 we can set a home-break (away-break) in period p if II If nbp,− Vk and only if the existing one is an away-break (home-break). → = 0 if and III We can set a break in period p ∈ {2, . . . , n − 1}, nbp,− Vk $ $ $ n $ br only if $P− → \ {1}$ < 2 − 1. Vk

Rule I states that a break in the first period is possible if this specific − → break has not been set on Vk . Rule II takes care of the fact that in each period either two or no breaks are set. Hence, if exactly one break has − → been arranged in period p on Vk the complementary break is possible in p. Rule III decides whether a break can occur in a period where no − → break is arranged on Vk yet. Since there can be no more than n2 periods having breaks (including the first period) a break in a period having no break so far is possible if less than n2 − 1 periods (excluding the first period) have breaks already. Clearly, rules I and II cover “No Break Twice”. No Three Consecutive Breaks: As shown in Briskorn and Drexl [11] we can further restrict the set of possible breaks by incorporating a necessary condition from Miyashiro et al. [64]: Due to restriction ′ ′ (2.7) for each subset T ′ ⊂ T there must be exactly |T |(|T2 |−1) matches between teams of T ′ , see (2.25). Theorem 5.3. A break sequence containing three (circular) consecutive periods leads to an infeasible subtree. Proof. Let p be the first of three consecutive periods having breaks. In each period having a break there is a home-break and an away-break. We combine three HAPs having breaks in p, p + 1, and p + 2 such that the break’s venue in p is equal to the break’s venue in p + 2 and different from the break’s venue in p+1. The three teams corresponding to these three HAPs cannot play against each other in period p′ ∈ {1, . . . , p − 1} ∪ {p + 2, . . . , n − 1}. There can be exactly one match among these teams in periods p and p + 1. Therefore, there is a subset ′ ′ of teams which can play only |T |(|T2 |−1) − 1 times against each other and which, consequently, violates (2.25). ⊓ ⊔ Table 5.2 provides an example of HAPs combined as done in the proof. According to theorem 5.3 we modify rule III of “Break Sequences” to rule III’.


89

Table 5.2. Example for 3 HAPs with too few matches Period

1

...

p−1

p

HAP 1 HAP 2 HAP 3

... ... ...

... ... ...

0 0 0

0 1 1

p+1 p+2 p+3 1 1 0

0 0 0

1 1 1

...

n−1

... ... ...

... ... ...

→ = 0 III’ We can set a break in period p ∈ {2, . . . , n − 1}, nbp,− Vk if a break is possible in ! period p according to III !and − → =0 ∧ − → = 0 ∨ nb − → = 0 ∧ nb → = 0 ∨ nb nbp−2,− p+1,Vk p−1,Vk Vk p−1,Vk !! → = 0 ∨ nb − → =0 nbp+1,− . V p+2,V k

k

Rule III’ checks whether a sequence of three consecutive break periods would be arranged if a break is fixed in period p. Note that we can further slightly strengthen III’ as III”.1 to III”.5 by taking into account that there must be two breaks in the first period in the final HAP set − → no matter whether they are already set on Vk or not.

→ = 0 if a III”.1 We can set a break in period p ∈ {4, . . . , n − 4}, nbp,− Vk break is possible in period p according to III’. → = 0 III”.2 We can set a break in period p = 3 if nb3,− Vk and if a break is possible in period !! 3 according to III and ! → = 0 ∧ nb − → = 0 ∨ nb − → =0 nb2,− . Vk 4,Vk 5,Vk → = 0 III”.3 We can set a break in period p = 2 if nb2,− Vk and if a break is possible ! in period 2 according to III and − → − → nbn−1,V = 0 ∧ nb3,V = 0 . k k → = 0 III”.4 We can set a break in period p = n − 1 if nbn−1,− Vk and if a break is possible! in period n − 1 according to III and → = 0 ∧ nb − → =0 . nbn−2,− Vk 2,Vk → = 0 III”.5 We can set a break in period p = n − 2 if nbn−2,− Vk and if a break!is possible in period n − 2 according to III and !! − → − → − → nbn−1,V = 0 ∧ nbn−3,V = 0 ∨ nbn−4,V = 0 . k

k

k

Rule III”.1 directly corresponds to III’ for p ∈ {4, . . . , n − 4}. Special cases are periods 3, 2, n − 1, and n − 2 in rules III”.2 to III”.5 being strengthened in comparison to III’. Here, the first period is not checked for breaks because in a complete HAP set with the minimum number of breaks there are exactly two breaks in it.

90


Feasible Sequence: In the following we generalize the

basic idea of “No Three Consecutive Breaks”. Choosing l ∈ 1, . . . , n2 break periods means fixing 2l HAPs. In order to check (2.25) we have to take care of 22l − 2l − 1 subsets of HAPs. Note that |T ′ | = 2 and |T ′ | = 3 are checked inherently by “No Break Twice” and “No Three Consecutive Breaks”, respectively. Here, we propose an efficient way to check all T ′ having exactly l HAPs and, therefore, 2ll subsets of HAPs. In order to establish a common notation we first introduce parts of the one in Miyashiro et al. [64]. We sort the given 2l HAPs as follows: We arrange two blocks of l HAPs each. The first (second) one consists of HAPs having 0 (1) in the last slot. Both blocks are ordered by ascending HAPs’ break periods. An example with 2l = 6 HAPs is shown on the left hand of table 5.3. Table 5.3. Example for ordered HAP set and equivalent representation period 1

2

3

4

5

period 1

2

3

4

5

HAP HAP HAP HAP HAP HAP

1 0 0 0 1 1

0 0 1 1 1 0

1 1 1 0 0 0

0 0 0 1 1 1

HAP HAP HAP HAP HAP HAP

1 0 0 0 1 1

1 1 0 0 0 1

1 1 1 0 0 0

1 1 1 0 0 0

1 2 3 4 5 6

0 1 1 1 0 0

1 2 3 4 5 6

1 0 0 0 1 1

The right hand side of table 5.3 provides an equivalent representation of the ordered (partial) HAP set as proposed in Miyashiro et al. [64]. The construction of this representation is done as follows. • For HAPs having 0 in the last slot set all entries ahead of the break period to 0. Set the entry in the break period and all entries behind to 1. • For HAPs having 1 in the last slot set all entries ahead of the break period to 1. Set the entry in the break period and all entries behind to 0. Note that min{c0 (T ′ , p), c1 (T ′ , p)} is equivalent in corresponding columns p for each ordered set of HAPs T ′ and its equivalent representation, see Miyashiro et al. [64]. As special cases of T ′ Miyashiro et al. [64] introduce cyclically consecutive sets of HAPs and narrow sets of HAPs. A set of HAPs T ′ is


91

narrow if and only if there is at least one period where all HAPs’ entries are identical. Miyashiro et al. [64] show that given a HAP set for each subset of HAPs T ′ which is not narrow there is a narrow subset of HAPs T ′′ such that (2.25) is at least as tight for T ′′ as it is for T ′ . Consequently, checking (2.25) is restricted to narrow subsets of a HAP set. Let T l be the set of subsets of HAPs having exactly l HAPs and ′ let T l ⊂ T l be the set of narrow subsets of HAPs having exactly one HAP for each break period. ′

Theorem 5.4. For each T ′ ∈ T l there is a T ′′ ∈ T l such that (2.25) is at least as tight for T ′′ as it is for T ′ . ′

′

Proof. Given an arbitrary T ′ ∈ T l \ T l we construct T ′′ ∈ T l such that (2.25) is at least as tight for T ′′ as it is for T ′ . Circulate columns of T ′ such that there is no break in the first period. Consider the equivalent representation of T ′ as shown on the right hand side of table 5.3. Then, c0 (T ′ , 1) = c1 (T ′ , n − 1) and c1 (T ′ , 1) = c0 (T ′ , n − 1). If l is even there is at least one break period p such that c0 (T ′ , p − 1) − 1 = c0 (T ′ , p) = 2l or c0 (T ′ , p − 1) + 1 = c0 (T ′ , p) = 2l . If l is odd there is at least one break period p such that min {c0 (T ′ , p − 1), c1 (T ′ , p − 1)} =

min {c0 (T ′ , p), c1 (T ′ , p)} = 2l . Now, we construct T ′′ as follows:

• If l is even set 2l ones and 2l zeros in period p. Additionally, set 2l + 1 ones and 2l − 1 zeros in period p − 1 by copying the pattern of p

% & and exchanging one 0 by 1. If l is odd set 2l zeros and 2l ones in % &

p. Additionally, set 2l ones and 2l zeros in p − 1 by copying the pattern of p and exchanging one 0 by 1. • Going backward from p − 1 set a break in break period p′ for an arbitrary HAP i having 0 in p′ (then, i’s entry in each period p′′ ∈ {1, . . . , p′ − 1} is 1). Thus, the number of zeros is decreased by 1 at the predecessor of each break period. Proceed until the first period is reached or there is no 0 left in any HAP. If there is no 0 left in break period p′ proceed by setting a break for an arbitrary HAP j not having a break yet in each break period (then, j’s entry in each period p′′ ∈ {1, . . . , p′ − 1} is 0). Thus, the number of zeros is increased by 1 at the predecessor of each break period p′′ ∈ {1, . . . , p′ − 1}. • Going forward from p set a break in break period p′ for an arbitrary HAP i having 1 in p′ − 1 (then, i’s entry in each period p′′ ∈ {1, . . . , p′ − 1} is 0). Thus, the number of zeros is increased by 1 in each break period. Proceed until the last period is reached or

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there is no 1 left in any HAP. If there is no 1 left in break period p′ proceed by setting a break for an arbitrary HAP j not having a break yet in each break period (then, j’s entry in each period p′′ ∈ {1, . . . , p′ − 1} is 1). Thus, the number of zeros is decreased by 1 in each break period p′′ ∈ {p′ , . . . , n − 1}. ′

First, we show that T ′′ ∈ T l . Obviously, T ′′ ∈ T l . Let op be the ordinal of p within the break sequence (after circulating). If op is greater or equal c0 (T ′ , p) then there is at least one period p′ ∈ {1, . . . , p − 1} having no 0. If op is lower or equal c0 (T ′ , p) then l − op is greater or equal than c1 (T ′ , p). Therefore, there is at least one period p′ ∈ ′ {p + 1, . . . , n − 1} having no 1. Thus, T ′′ ∈ T l . Second, we show that (2.25) is at least as tight for T ′′ as it is for T ′ . Obviously, due to construction

min c0 (T ′ , p), c1 (T ′ , p) = min c0 (T ′′ , p), c1 (T ′′ , p) .

Note that, depending on T ′ the minimization term of (2.25) is increased by 1, is decreased by 1, or is not changed at all at each break period. The latter case appears at break period p′ if two complementary HAPs with breaks in p′ are contained in T ′ . T ′′ does not contain any complementary HAPs by construction and, therefore, the minimization term of (2.25) is increased by 1 or is decreased by 1, respectively, at each break period. Hence,

min c0 (T ′ , p + k), c1 (T ′ , p + k) ≥ min c0 (T ′′ , p + k), c1 (T ′′ , p + k) n−2 n−2 , . . . , −1 ∪ 1, . . . , ∀k ∈ − 2 2

(indices taken modulo n − 1) since going backward and forward from p, respectively, the number of zeros or ones is strictly lowered to zero in T ′′ at each break period. ⊓ ⊔ In order to illustrate the construction as done in the proof above we provide tables 5.4 to 5.6. A HAP set (and, therefore, a subset of HAPs) T ′ is given on the left hand side of table 5.4. The HAPs’ break periods differ from each other ′ and, moreover, T ′ is not narrow. Therefore, T ′ ∈ T l \ T l with l = 4. First, we circulate periods until we have no break in the first period. This results into the set of HAPs shown on the right hand side of table 5.4 and does not affect the sum in term (2.25). The left hand side of table 5.5 shows the equivalent representation of T ′ . We choose break period p = 2 since c0 (T ′ , 2− 1)+ 1 = c0 (T ′ , 2) = 2l .


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Table 5.4. T ′ before (left) and after (right) circulating Period 1 2 3 4 5 6 7

Period 1 2 3 4 5 6 7

HAP HAP HAP HAP

HAP HAP HAP HAP

1 2 3 4

0 0 0 1

0 1 1 0

10 01 01 11

10 10 01 01

1 1 0 0

Min(0,1) 1 2 2 1 2 2 2

1 2 3 4

1 1 0 0

00 01 01 10

1 0 0 1

01 11 10 10

0 0 1 1

Min(0,1) 2 1 2 2 1 2 2

Table 5.5. Equivalent representation of T ′ (left) and T ′′ (right) Period 7 1 2 3 4 5 6

Period 7 1 2 3 4 5 6

HAP HAP HAP HAP

HAP HAP HAP HAP

1 2 3 4

0 0 1 1

0 0 0 1

11 00 00 11

11 01 00 00

1 1 0 0

Min(0,1) 2 1 2 2 1 2 2

1 2 3 4

1 1 1 1

11 10 00 11

1 0 0 1

00 00 00 10

0 0 0 0

Min(0,1) 0 1 2 2 1 0 0

Then, we construct T ′′ by reducing the number of zeros at each break period going backward from p to the first period and reducing the number of ones at each break period going forward from p to the last period. The result is shown on the right hand side of table 5.5. Note that min {c0 (T ′ , 2), c1 (T ′ , 2)} = min {c0 (T ′′ , 2), c1 (T ′′ , 2)}. Since the minimization term’s value in (2.25) is strictly reduced at each break period going backward and forward from p = 2 in T ′′ min {c0 (T ′ , p), c1 (T ′ , p)} ≥ min {c0 (T ′′ , p), c1 (T ′′ , p)} for each p ∈ P . Table 5.6. T ′′ before (left) and after (right) recirculating Period 7 1 2 3 4 5 6

Period 7 1 2 3 4 5 6

HAP HAP HAP HAP

HAP HAP HAP HAP

1 2 3 4

1 1 1 1

0 0 1 0

10 01 01 10

01 01 01 11

0 0 0 0

Min(0,1) 0 1 2 2 1 0 0

1 2 3 4

0 0 1 0

1 0 0 1

00 10 10 01

10 10 10 10

1 1 1 1

Min(0,1) 1 2 2 1 0 0 0

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On the left hand side of table 5.6 we illustrate the interpretation of the equivalent representation of T ′′ as a subset of HAPs. On the right hand side the set of HAPs is recirculated which finishes the construction of T ′′ . Thus, given a set of l break periods we can check condition (2.25) for ′ each T ′ ∈ T l by checking condition (2.25) for each T ′′ ∈ T l according 'l l+k l to theorem 5.4. Note that |T l | = 2ll = (2l)! k=1 k > 2 if l > 1 l!·l! = ′ l and |T | = 2l which makes this reduction extremely useful. ′ Since each T ∈ T l is narrow there exists a period pT such that from pT to the following break period all HAPs of T have identical entries. ′ Therefore, T ∈ T l can be specified by pT and the singleton entry ′ in these periods. We can further reduce T l by eliminating subsets of HAPs being complementary to each other: condition (2.25) is equally ′ ′ tight for two subsets of HAPs T1 ∈ T l and T2 ∈ T l with pT1 = pT2 and different entries in period pT1 as shown in Miyashiro et al. [64]. Therefore, we check only the l subsets of HAPs having entry zero in periods where all entries are identical. As shown in Miyashiro et al. [65] each check can be done in linear time. The test above is applied in addition to “No Three Consecutive Breaks”. We apply it only for l ≥ 4 since “No Three Consecutive Breaks” covers l < 4. Feasible Subsequences: Note that T l ⊂ 2T . Hence, checking condition (2.25) for each T ′ ∈ T l as proposed by “Feasible Sequence” can not ensure condition (2.25) for each T ′ ⊆ T . For an example consider the partial HAP set h illustrated in table 5.7. Table 5.7. Infeasible subsequence period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 HAP HAP HAP HAP

1 2 3 4

010 010 010 010

10 10 10 10

01 10 10 10

01 11 10 10

0 0 0 1

1 1 1 1

0 0 0 0

1 1 1 1

0 0 0 0

1 1 1 1

Checking break sequence {6, 9, 10, 11} according to “Feasible Sequence” does not identify h to be infeasible. Clearly, break sequence {9, 10, 11} would be identified as infeasible but is never checked according to “Feasible Sequence” if p = 11 is the break period being added last (in the course of our branching scheme). In order to fix this flaw we propose to check subsequences of break sequences, as well.


95

According to Miyashiro et al. [64] not each subset of HAPs must be checked if the minimum number of breaks is required. We have to check only those subsets having no more than n2 HAPs and exclusively containing cyclically consecutive HAPs. We aim at implicitly checking all these subsets of HAPs by checking subsequences. We need to check only cyclically consecutive subsequences of break periods since subsequences not being cyclically consecutive correspond to sets of HAPs not being cyclically consecutive. Note that there exist exactly two narrow and cyclically consecutive sets of HAPs corresponding to a given cyclically consecutive subsequence of break periods. These two sets of HAPs are complementary to each other and, hence, we have to check only one set of HAPs for each subsequence. According to our branching scheme we add break periods step by step. Therefore, we can restrict checking subsequences to subsequences incorporating the new break period in each step. Subsequences not containing the new break period have been checked before. Again, checking a specific (sub-)sequence of break periods is done according to Miyashiro et al. [65] in linear time. 5.3.2 Choice of Branching Candidates Since the branching scheme prescribes to create child nodes corresponding to each possible break of a specific team branching candidates correspond to teams (namely: branching teams), here. We propose several strategies to choose the branching team below. First, we define a frac′ = |1 − tional break value bri,p j∈T \{i} (xi,j,p−1 + xi,j,p ) | where xi,j,0 means xi,j,n−1 for each team i ∈ T and period p ∈ P according to the current node’s optimal solution. Random Choice: Choosing the branching team randomly minimizes computational effort. Here, we implement a (pseudo-)random selection by choosing the team having index dk + 1 where dk is the depth of the current node k. = Largest Fractional Break: We choose team i′ ′ ′ with largest fractional break arg maxi maxp bri,p | bri,p < 1 value as branching team. The motivation for this strategy is as follows. First, the current node’s optimal solution is cut by fixing i′ ’s break. Second, i′ having this specific break is likely to enable low cost tournaments. Most Infeasible 1: Among all teams we choose the one having the most infeasible constellation of break values. The common idea is to enforce feasibility for those teams first where least tendency is given

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in which period to set their break, see Achterberg et al. [1]. Note that not each team must have at least one break (including those in the first period) in a feasible solution to the LP relaxation. Accordingly, afeasible solution to the LP relaxation might provide a team i with ′ the structure of p∈P bri,p > 1. Since this effect severely contradicts ′ ′ as br a feasible IP solution we choose team i = arg maxi i,p p∈P ′ branching team if p∈P bri′ ,p > 1. Otherwise, we choose the branching team according to “Largest Fractional Break”. ′ } . Most Infeasible 2: We choose team i′ = arg mini maxp {bri,p The basic idea is the same as for “Most Infeasible 1” but infeasibility is measured differently. There can be two reasons for the maximum fractional break value being small. First, p∈P bri′′ ,p is small (in particular ′ ′ p∈P bri′ ,p = 1 but fractional break values p∈P bri′ ,p < 1). Second, bri′′ ,p are larger than 0 for many periods p. Both effects contradict the structure of a feasible IP solution. 5.3.3 Node Order Strategy Just as described in section 5.2.3 we consider “Depth First Search” and “Breadth First Search” as node order strategies. We observe that “Breadth First Search” is significantly more efficient due to less nodes being explored for instances with less than 10 teams. For larger instances our list of nodes grows that large that administrative effort uses up this advantage. Therefore, we propose a compromise between a short node list guaranteed by “Depth First Search” and a small number of nodes being explored guaranteed by “Breadth First Search” based on the father node’s optimal LP solution. First, we obtain vcur as normalized current node’s solution value dividing it by the lower bound. Second, we calculate the fraction xbin of match variables having binary values in the current node’s optimal solution. Then, we sort the node list in ascending order of vcur − w · xbin where w ∈ R. The idea is to explore a node earlier if its father’s optimal solution provides many binary variables. In preliminary tests w = 1.5 proofed to be a good choice as far as running times are concerned. According to the strategies proposed above all children of a single node are considered identical. Therefore, we additionally have to decide in which order nodes having the same father are explored. We propose several strategies hereafter.


97

Pseudo Cost: We calculate pseudo cost chi,p and cai,p representing average cost of matches being possible if i is branched to have a homebreak or away-break, respectively, in period p.

chi,p =

1 (n − 1)2 1 (n − 1)2

cai,p =

1 (n − 1)2 1 (n − 1)2



 p−1 p−1 ⌋ ⌉ ⌈ ⌊ 2 2   cj,i,p−2p′ + ci,j,p+1−2p′  + 

j∈T \{i}



 

j∈T \{i}



p′ =1

n−p ⌈ 2 ⌉

p′ =1

ci,j,p−2+2p′ +

p′ =1

n−p ⌊ 2 ⌋

p′ =1



 cj,i,p−1+2p′ (5.2)

 p−1 p−1 ⌋ ⌉ ⌈ ⌊ 2 2   ci,j,p−2p′ + cj,i,p+1−2p′  + 

j∈T \{i}



 

j∈T \{i}

p′ =1

n−p ⌈ 2 ⌉

p′ =1

p′ =1

cj,i,p−2+2p′ +

n−p ⌊ 2 ⌋

p′ =1



 ci,j,p−1+2p′ (5.3)

Child nodes are explored in ascending order of pseudo cost chi,p and corresponding to breaks at home and away, respectively, of branching team i in period p. Since chi,p and cai,p are static we calculate them once before the B&B procedure starts and, therefore, computational effort is small. Pseudo Cost Revisited: If we branch on team i to have a specific break pseudo costs according to other teams may be altered in the resulting branching subtree. Suppose, for example, team i is branched to have a home-break in period 2 and j is chosen as branching team in the corresponding subtree. In pseudo-cost caj,4 cost cj,i,2 and ci,j,3 should not be considered (in contrast to calculation in (5.3)) since these matches are not possible due to the fixed break of i. More generally, suppose team i is branched to have a break in period p. Additionally, a break for branching team j in period p′ is considered in the subtree. We eliminate cai,p

• cost according to a match between teams i and j in period p′′ ∈ {min{p, p′ }, . . . , max{p, p′ } − 1} from pseudo cost according to j and period p′ if either i’s and j’s break venues are identical and |p − p′ | is odd or the break venues differ and |p − p′ | is even,

98


• cost according to a match between teams i and j in period p′′ ∈ {1, . . . , min{p, p′ } − 1} ∪ {max{p, p′ }, . . . , n − 1} from pseudo cost according to j and period p′ if either i’s and j’s break venues are identical and |p − p′ | is even or the break venues differ and |p − p′ | is odd. Then, child nodes are explored in ascending order of revisited pseudo cost. Largest Fractional Break First: We first explore the child node corresponding to the branching team i’s break (defined by period and ′ according venue) which implies the largest fractional break value bri,p to the father node’s optimal solution. The remaining child nodes are explored in ascending order of “Pseudo Cost” or “Pseudo Cost Revisited”, respectively. The idea is here that a large fractional break value might indicate a profitable break due to the LP problem’s objective of cost minimization.

5.4 Computational Results In this section we line out efficiency of the branching schemes proposed in sections 5.2 and 5.3. For both we present the configuration proofed to be the most efficient one. Beside run times we line out both schemes’ efficiency as far as avoiding infeasible HAP sets is concerned. 5.4.1 General Home-Away-Pattern Sets Results are based on the formulation of the single RRT problem presented in section 2.1, namely SRRTP-IP. Among the strategies to choose the branching candidate “Most Fractional” turned out to be the most efficient one. Additionally, we employ “Breadth First Search” as node order strategy. In table 5.8 we line out the average number of LP problems solved (LP), the average number of LP problems being infeasible due to the HAP set (i.LP), the average number of IP problems solved since branching on each team’s venue in each period does not provide integer solutions (IP), average run times of our approach (r.t.), and average run times needed by Cplex (r.t. Cplex) as given before in table 3.5. Run times indicate that our branching scheme can not compete with Cplex when applied to the single RRT problem. However, according to theorems 5.1 and 5.2 the choice of branching candidates takes implicitly care of several necessary conditions for HAP

5.4 Computational Results

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Table 5.8. Comp. Results for Branching on General HAP Sets n

LP i.LP IP

6 3.4 8 15.7 170.3 10 12 4124.9 14 136627.1

0 0 0 0 0

r.t. r.t. Cplex

0 0.01 0 0.03 0 0.70 0 54.02 0 4602.29

0.01 0.05 0.37 2.03 34.43

sets to be feasible. This is impressively confirmed by our test runs. Each single (partial) HAP set corresponds to a feasible LP problem. Note that this does not necessarily imply feasibility of each (partial) HAP set. Moreover, not a single IP problem is solved since each branching path was pruned beforehand. Hence, although our branching scheme can not guarantee integer solutions probability is high that it suffices to reach optimality. Considering these aspects it seems advisable to employ it in the approach developed in chapter 6. 5.4.2 Minimum Number of Breaks Here, results are based on the model formulation given in section 3.2.1. First, we evaluate our strategies to avoid infeasible HAP sets. We choose branching candidates according to “Random Choice”. Furthermore, we employ depth first search and insert child nodes in ascending order of the corresponding breaks’ periods. Since the resulting branching scheme is static and has no cost orientation, differences in the number of nodes being explored as well as run times are exclusively caused by decisions whether a node corresponding to a specific break is constructed or not. Strategies “No Restrictions”, “No Break Twice”, “Break Sequences”, and “No Three Consecutive Breaks” are carried out for 8 teams since the run times grow too high for instances with more teams as long as no efficient strategy is employed. Results are given in relation to results obtained using “No Restrictions”. Evaluation of strategy “Feasible Sequence” as described in section 5.3.1 and a special case, namely “Four Breaks in Five Periods”, is done solving instances having 10 teams. Here, results are given in relation to results obtained using “No Three Consecutive Breaks” for n = 10. Each strategy is employed in addition to the previous ones except “Four Breaks in Five Periods” replacing “Feasible Sequence”.

100


We focus on the decreasing number of LP problems to be solved (LP red.) and infeasible LP problems (i.LP red.), respectively. Furthermore, the reduction of average run times (r.t. red.) is given in table 5.9. Table 5.9. Comp. Results for Branching Strategies Strategy No Restrictions (n = 8) No Break Twice (n = 8) Break Sequences (n = 8) No Three Consecutive Breaks (n = 8) No Three Consecutive Breaks (n = 10) Feasible Sequence (n = 10) Four Breaks in Five Periods (n = 10)

LP red. i.LP red. r.t. red. — 26.1% 49.4%

— 48.7% 87.3%

— 6.7% 22.9%

72.7%

100.0%

54.6%

— 30.0%

— 99.5%

— 28.9%

33.0%

100.0%

32.1%

As we can see the number of nodes being explored is strictly decreased by each strategy. Employing “No Break Twice” and “Break Sequences” reduces the number of LP problems by 50%. More remarkably, the number of infeasible LP problems is reduced by more than 85%. Note that both strategies avoid constructing infeasible (partial) HAPs which would immediately result into an infeasible LP problem. “No Three Consecutive Breaks” takes a step further by avoiding infeasible (partial) HAPs not corresponding to an infeasible LP problem but corresponding to an infeasible subtree. Here, we omit solving these LP problems and, possibly, further child nodes. Applying “No Three Consecutive Breaks” we do not obtain a single infeasible LP problem. The overall number of LP problems and run times are reduced by more than 70% and more than 50%, respectively. Considering 10 teams nearly each infeasible LP problem is avoided if “Feasible Sequence” is applied. Here, we recognize a special case for 10 teams. Four break periods in a sequence of five periods (without three consecutive break periods) is not infeasible in general but for n = 10. The fifth break period can not be feasibly chosen according to “Feasible Sequence”. In order to fine-tune our approach we consider strategy “Four Breaks in Five Periods” and observe that no infeasible LP problem remains to be solved. Both the overall number of LP problems and run times are reduced by more than 30%. Note that “Feasible Subse-

5.5 Summary

101

quences” is not evaluated, here, since this strategy takes effect only for problems with more than 10 teams. However, tests for larger problems are not possible due to run times. For the sake of completeness we line out that the average number of IP problems which are solved for each problem instance is 0.9 for n = 8 and 4.7 for n = 10. Among strategies to choose the branching candidate “Most Infeasible 1” turned out to be the most efficient. Additionally, we employ “Breadth First Search” considering the number of binary variables and “Pseudo Cost Revisited” as node order strategies. Out of the strategies evaluated above “No Three Consecutive Breaks” combined with “Four Breaks in Five Periods” is applied here. Run times are given in table 5.10. Table 5.10. Comp. Results for Minimum Number of Breaks n

B&B Cplex

6 0.15 0.23 8 12.62 26.79 10 7841.14 —

Clearly, our B&B approach outperforms Cplex for n < 10. Cplex runs out of memory after about 12 hours for n = 10. Note that Cplex was employed using “Depth First Search” to overcome lack of memory in Briskorn and Drexl [10, 11]. Not even a feasible solution is found within 6 days of running time, then. Accordingly, proofing optimality in 2.2 hours on average clearly indicates superiority of our approach.

5.5 Summary We propose a B&B approach in order to find minimum cost RRT schedules. The basic idea originates from a decomposition scheme fixing each team’s venue in each period, first, and arranging matches, afterwards. We distinguish between the general case and RRT schedules with the minimum number of breaks. Results are twofold for the general case. The branching scheme suffices to obtain optimal solutions and to avoid infeasible subtrees. However, run times are significant larger than those needed by Cplex. For the minimum number of breaks our branching scheme clearly outperforms Cplex. An outstanding difference between both cases might be

102


the fact that we are able to prune whole subtrees for the minimum number of breaks while we can prune only single nodes in the general case. Since both branching schemes can not guarantee integer solutions we have to handle the case where no branching candidate is given for a fractional solution. We propose to solve the corresponding IP problem using Cplex. Since solving IP problems is significantly more time consuming than solving LP problems we emphasize that for all test instances only a very small number of IP problems had to be solved. In fact, no IP problem was solved in the general case. Our branching schemes are likely to obtain HAP sets that are feasible. Consequently, most probably only IP problems corresponding to feasible HAP sets have to be solved.

6 Branch–and–Price Algorithm

6.1 Motivation and Basic Idea In this chapter we aim at handling problems observed in chapters 2 and 3. When we tackle RRT problems with standard B&B methods we suffer from four main obstacles so far: • The problem size is determined by more than n(n − 1)2 variables constraints and allows exact solution only and more than 3n(n−1) 2 for rather small instances. • Solutions to LP relaxations in general are highly fractional and provide poor lower bounds. • When setting variables to integer values in a B&B framework we often experience fractional values coming up for other variables due to model inherent symmetry. • Cost oriented node order strategies are difficult to implement since fixing variables has intractable consequences for other variables due to the compact structure of time constrained SLS problems. Consequently, we propose a reformulation for SRRT-IP introduced in section 2.1 and further constraints presented in chapter 3. We expect less symmetry and fewer constraints when employing more meaningful columns. Moreover, we aim at tightening the provided lower bound. This step is motivated by, besides others, Mehrotra and Trick [60] who faced similar problems by solving the problem to color the vertices of a graph with the minimum number of colors. Since the number of variables of the reformulation is exponential (in n) we introduce a column generation (CG) model to be employed in a branch-and-price framework in order to enforce integrality.

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6 Branch–and–Price

6.2 Reformulation Everything below is based on SRRT-IP and its extensions. However, adaptation to other basic problems introduced in section 2.2 is straightforward. We define a MD as a set of matches where each team i ∈ T plays exactly once. Columns and master’s variables, respectively, correspond to MDs being assigned to a specific period, scheduled MDs namely. The set of all scheduled MDs is denoted by P M . We say a pair (i, j), i, j ∈ T , i = j, of teams is contained in m ∈ P M (denoted by (i, j) ∈ m) if m comprises a match between teams i and j. Additionally, an ordered pair − → − → (i, j), i, j ∈ T , i = j, is contained in m ∈ P M (denoted by (i, j) ∈ m) if m comprises a match of team i at home against team j. Cardinality of P M is given in (6.1). |P M | = (n − 1)

n * n! n ! n2 . = (n − 1) i > n 2 n 2!

(6.1)

i= 2 +1

We denote the period of m ∈ P M by p(m). Cost cm of scheduled MD m ∈ P M is equal to the sum of the costs of all ordered pairings contained in m if carried out in period p(m) as outlined in (6.2). cm = ci,j,p(m) ∀m ∈ P M (6.2) −−→ (i,j)∈m

In the following we present a master problem and the corresponding pricing problem. We use several constraints given in chapters 2 and 3. Thereby, we distinguish between constraints relating to a single MD only and constraints involving interrelations between several MDs. The first class of constraints is used in the pricing problem while the second class is considered in the master problem. 6.2.1 Set Partitioning Master Problem We propose an integer programming model equivalent to SRRT-IP employing binary variables ym , m ∈ P M , being equal to 1 if and only if scheduled MD m is chosen for a single RRT. Objective function (6.3) represents the goal of finding the minimum cost tournament. Constraint (6.4) forces each pair of teams to meet exactly once and it is equivalent to (2.7). Equality (6.5) assures exactly one MD being arranged in each period of the tournament and it corresponds to (2.8) since each team i ∈ T participates in each scheduled MD m ∈ P M exactly once.

6.2 Reformulation Model 6.1: CG Master – SRRT cm y m min

105

(6.3)

m∈P M

s.t.

ym

=

1

∀ i, j ∈ T, i < j

(6.4)

ym

=

1

∀p∈P

(6.5)

∈

{0, 1} ∀ m ∈ P M

m∈P M,(i,j)∈m

m∈P M,p(m)=p

ym

(6.6)

As mentioned above the number of variables is exponential in the number of teams. Therefore, it is reasonable to think of the LP relaxation to CG Master – SRRT as a restricted master problem in a CG process initialized with a subset of columns. Iteratively, columns are generated according to the current restricted master’s optimal solution in order to reach the master’s optimal solution. Consequently, we consider the LP relaxation to CG Master – SRRT in the following. As shown in de Carvalho [18] the CG approach can be accelerated if the domain of dual variables is restricted by restricting solution space of the master problem. Obviously, either (6.4) or (6.5) can be relaxed to “no less than” constraints and “no more than” constraints, respectively. Solving the resulting problem will still lead to solutions fulfilling both restrictions with equality. Relaxing both constraints to “no more than” constraints is not possible even if cm is not greater than zero for each m ∈ P M . Rosa and Wallis [71] proof that there are premature sets of 1-factors. This means that it is possible to choose a set of 1-factors having no edge in common but not being part of any 1-factorization. Therefore, the goal of cost minimization can not guarantee constraints (6.4) and (6.5) being obeyed with equality. Analogously, relaxing both restrictions to “no less than” constraints is not possible either. Instead, relaxing one of them to a “no more than” constraint while relaxing the other one to be a “no less than” constraint provides feasible solutions to CG Master – SRRT. We evaluated all variants and conclude that the relaxation having both constraints set to “no more than” constraints leads to shortest solution times but can not guarantee a feasible solution for CG Master – SRRT. In order to increase the probability for feasible solutions we

106


transform all ci,j,p to have values not greater than zero by subtracting the maximum cost value as outlined in (6.7). Obviously, this has no effect on the optimal solution’s structure. ci,j,p = ci,j,p −

max

i′ ,j ′ ∈T,i′ =j ′ ,p′ ∈P

{ci′ ,j ′ ,p′ }

∀i, j ∈ T, i = j, p ∈ P (6.7)

Among all feasible relaxations the variant having constraint (6.4) relaxed to “no more than” constraint and (6.5) kept unchanged provides lowest run times. Consequently, in the remainder we will restrict ourselves to these two variants. The LP relaxation of CG Master – SRRT provides a lower bound to the original problem which is not lower than the one provided by the LP relaxation. However, it might be larger and, therefore, more useful. Clearly, each solution to the LP relaxation of CG Master – SRRT is a solution to the LP relaxation of SRRT-IP. On the other hand, there are solutions to the LP relaxation of SRRT-IP being not feasible to the LP relaxation of CG Master – SRRT. As pointed out in Trick [82] the lower bound given by the LP relaxation of SRRT-IP can be strengthened by adding odd set constraints as shown in (6.8).

(xi,j,p + xj,i,p)

≥

∀ p ∈ P, T ′ ⊂ T, |T ′ | odd (6.8)

1

i∈T ′ j∈T \T ′

Theorem 6.1. Solutions to the LP relaxation of CG Master – SRRT fulfill all odd set constraints (6.8). Proof. A solution to the LP relaxation of CG Master – SRRT corresponds to a convex combination of scheduled MDs in P Mp = {m | m ∈ P M, p(m) = p} for each period p. Trivially, each scheduled MD m ∈ P M fulfills (6.8). Let o(T ′ , p) be the value of matches between teams in T ′ and teams in T \ T ′ in period p.

o(T ′ , p) =

m∈P M,p(m)=p

=

i∈T ′

ym

m∈P M,p(m)=p

=1

ym

j∈T \T ′ ,(i,j)∈m

i∈T ′ j∈T \T ′ ,(i,j)∈m

m∈P M,p(m)=p

≥

ym

1

6.2 Reformulation

107

⊓ ⊔ Thus, we obtain a lower bound being not lower than the one of SRRTIP after adding each odd set constraint. Beside the constraints assuring a single RRT we use two of those restrictions introduced in chapter 3 considering more than one MD: minimum number of breaks and changing opponents’ strengths. In order to consider breaks we employ binary variables bri,p as introduced in chapter 3. Constraints (6.9), (6.10), (6.11), and (6.12) are equivalent to (3.5), (3.6), (3.7), and (3.8).

Model 6.2: CG Master – Minimum Number of Breaks ym − bri,p ≤ 1 ∀ i ∈ T, p ∈ P ≥2

(6.9)

→ j∈T,j =i m∈P M,(− i,j)∈m,

p(m)∈{p−1,p}

ym − bri,p

∀ i ∈ T, p ∈ P ≥2

≤

1

bri,p

≤

n−2

bri,p

∈ {0, 1} ∀ i ∈ T, p ∈ P ≥2

(6.10)

→ j∈T,j =i m∈P M,(− j,i)∈m, p(m)∈{p−1,p}

(6.11)

i∈T p∈P ≥2

(6.12)

In order to consider changing opponents’ strengths we employ binary variables esci,p known from chapter 3. Recall that S denotes the set of strength groups and the number of violations of the changing opponents’ strengths postulation is limited by escmax . Constraints (6.13), (6.14), and (6.15) correspond to (3.13), (3.14), and (3.15). We use integer variable esvio in order to allow solutions having more than escmax violations of changing opponents’ strengths restriction for specific team i ∈ T . The necessityfor this is outlined in section 6.4.2 in detail. We associate cost Mvio = i∈T j∈T \{i} p∈P ci,j,p with esvio to punish violations and obtain objective function (6.3a): min cm ym + Mvio esvio (6.3a) m∈P M

Further concepts from chapter 3 considering relations between more than one MD can be easily handled analogously. However, we choose the two above since they are among the most prominent ones.

108


Model 6.3: CG Master – Changing Strength Groups ym − esci,p ≤ 1 ∀ i ∈ T, S ′ ∈ S, p ∈ P ≥2 j∈S ′ ,j =i m∈P M,(i,j)∈m,p−1≤p(m)≤p

(6.13)

esci,p

− esvio

≤

escmax ∀i

esci,p

∈

{0, 1} ∀i ∈ T, p ∈ P ≥2

∈T

(6.14)

p∈P ≥2

(6.15) esvio

∈

N0

(6.16)

6.2.2 Matching Subproblem According to the well known concept of CG developed in Gilmore and Gomory [45] the subproblem is to find the column, i.e. MD assigned to a specific period, having the lowest reduced cost. Reduced cost cm of m ∈ P M is based on original cost cm and dual variables provided by the optimal solution to the current restricted master problem as given in table 6.1. Dual variable βp is restricted to non-positive values if (6.5) is formulated as “no more than” constraint and is not restricted otherwise. Table 6.1. Dual Variables of CG Master eq.

dual variables

(6.4) (6.5) (6.9) (6.10) (6.11) (6.13) (6.14)

αi,j ≤ 0 ∀ i, j ∈ T, i < j βp ∈ R, βp ≤ 0 ∀ p ∈ P γi,p ≤ 0 ∀ i ∈ T, p ∈ P ≥2 δi,p ≤ 0 ∀ i ∈ T, p ∈ P ≥2 ǫ≤0 ζi,S ′ ,p ≤ 0 ∀ i ∈ T, S ′ ∈ S, p ∈ P ≥2 ηi ≤ 0 ∀i ∈ T

Reduced cost cm of MD m ∈ P M is defined according to terms (6.17) to (6.19).

6.2 Reformulation

cm = cm −

αi,j − βp(m)

(6.17)

(i,j)∈m,i<j

−



−



  γi,p(m) + δj,p(m) +γi,p(m)+1 + δj,p(m)+1  + ,.+ ,.

− → (i,j)∈m



if p(m)≥2

109

if p(m)≤|P |−1

(6.18) 

  ζi,S(j),p + ζj,S(i),p +ζi,S(j),p+1 + ζj,S(i),p+1  (6.19) + ,.+ ,.

− → (i,j)∈m

if p(m)≥2

if p(m)≤|P |−1

Dual variables corresponding to constraints involving matches (i, j, p) contained in m are subtracted from original cost cm . Term (6.17) incorporates dual variables corresponding to constraints of CG Master – SRRT. Term (6.18) employs dual variables corresponding to (6.9) and (6.10). If no breaks are considered term (6.18) is not incorporated. Term (6.19) involves dual variables corresponding to (6.13) and is ignored if changing opponents’ strengths is not required. Note that dual variables ǫ and ηi do not contribute to reduced cost since constraints (6.11) and (6.14) do not directly incorporate scheduled MDs. We break terms (6.17) to (6.19) down to the contribution / ci,j,p of each match (i, j, p) to reduced cost of a scheduled MD m ∈ P M with − → p(m) = p and (i, j) ∈ m as outlined in (6.20) to (6.22). We refer to ci,j,p as reduced cost of match (i, j, p) in the remainder. / 2βp + n −γi,p − δj,p −γi,p+1 − δj,p+1 + ,.+ ,. +

ci,j,p = ci,j,p − αmin{i,j},max{i,j} − / if p≥2

(6.21)

if p≤|P |−1

−ζi,S(j),p − ζj,S(i),p −ζi,S(j),p+1 − ζj,S(i),p+1 + ,.+ ,. if p≥2

(6.20)

(6.22)

if p≤|P |−1

Note, that βp is fixed for a specific period p and, hence, we can distribute it equally on all possible matches since exactly n2 of them will contribute to a scheduled MD. Using reduced cost / ci,j,p we represent our subproblem M D. We employ binary variables xi,j being equal to one if and only if the scheduled − → MD contains oriented pairing (i, j), binary variables y p being equal to

110


one if and only if the MD is assigned to period p, and continuous vari− → ables ci,j representing reduced cost of oriented pairing (i, j) according to the chosen period p.

Model 6.4: CG Subproblem – M D min ci,j xi,j

(6.23)

i∈T j∈T \{i}

s.t.

(xi,j + xj,i ) = 1

∀i ∈ T

(6.24)

∀i, j ∈ T : j = i, p ∈ P

(6.25)

∀i, j ∈ T : j = i, p ∈ P

(6.26)

j∈T :j =i

ci,j,p ≥ ci,j (1 − y p )M + / (y p − 1)M + / ci,j,p ≤ ci,j yp =1

(6.27)

p∈P

xi,j

∈ {0, 1} ∀i, j ∈ T : j = i

(6.28)

yp

∈ {0, 1} ∀p ∈ P

(6.29)

ci,j

∈R

(6.30)

∀i, j ∈ T : j = i

Objective function (6.23) is to minimize arranged matches reduced cost according to the chosen period. Constraint (6.24) assures that each team participates in exactly one pairing. Restrictions (6.25) and (6.26) ci,j,p according to the chosen period p with set cost ci,j to reduced cost / y p = 1. Note that M is set to maxi,j∈T,i=j,p∈P {|ci,j,p |}. Equation (6.27) assures that the MD is assigned to exactly one period. M D is a quadratic mixed IP problem. In order to reduce computational effort we decompose M D into |P | subproblems M Dp , p ∈ P . M Dp represents the goal to find the minimum reduced cost scheduled MD m with p(m) = p. Hence, M Dp is equivalent to M D with yp = 1. Consequently, M D can be represented as M D ′ . As mentioned before we include all restrictions corresponding to a single MD into the subproblem. Therefore, we obtain several constraints to be added to terms (6.23) to (6.28) if under consideration. Parameters are identical to those introduced in chapter 3. Constraints (6.35), (6.36), (6.37), and (6.38) directly correspond to (3.1), (3.2), (3.3), and (3.4).

6.3 Branching Scheme Model 6.5: CG Subproblem – M Dp ci,j,p xi,j / min zp =

111

(6.31)

i∈T j∈T \{i}

s.t.

(xi,j + xj,i ) = 1

∀i ∈ T

(6.32)

j∈T :j =i

∈ {0, 1}∀i, j : j = i

xi,j

(6.33)

Model 6.6: CG Subproblem – M D′ min zp

(6.34)

p∈P

Model 6.7: CG Subproblem – M Dpext xi,j

≤

πi,j,p

∀i, j ∈ T, i = j ′

(6.35)

xi,j

≤

CR′

∀R ∈ R

(6.36)

xj,i

≤

CR′

∀R′ ∈ R

(6.37)

ai,j xi,j

≤

amax

i∈R′ j∈T \{i}

i∈R′

j∈T \{i}

(6.38)

i∈T j∈T \{i}

6.3 Branching Scheme In the following two components of our branching scheme are discussed. First, we propose alternative ways to create child nodes after solving a node’s LP problem. Second, strategies to prioritize nodes are presented. 6.3.1 Branching Strategy The most common way to create subproblems in a branching framework is to divide a variable’s domain into two (or more) distinct subdomains. Unfortunately, several problems arise when doing so for column generation models, see Vanderbeck [84] for example. This implies the need for branching strategies not based on variables’ domains.

112


Home–Away–Pattern Based Branching Since the branching schemes developed in chapter 5 turned out to be capable of avoiding infeasible subproblems we employ them in the B&P approach as well. General Home-Away-Pattern Sets The branching scheme considered in section 5.2 is not restricted to a special class of HAP sets. The idea is to branch by forcing team i to play at home or away, respectively, in period p if i does not exclusively play at home or away in p according to the optimal solution to the LP relaxation. Implementation is done by fixing variables xj,i or xi,j , j ∈ T , j = i, to zero in pricing problem M Dp if i is branched to play at home or away, respectively, in p. Pricing problems M Dp′ , p′ = p, are not affected. Choice of branching candidates and node ordering is done according to sections 5.2.2 and 5.2.3, respectively. Minimum Number of Breaks The branching scheme considered in section 5.3 is restricted to HAP sets having the minimum number of breaks. Here, branching is done by forcing team i to have a specific break (defined by period and venue) if i does not have exactly one (binary) break in the current node’s optimal solution. Coherence of setting breaks and fixing variables to zero is outlined in section 5.3. Setting a break, consequently, affects pricing problem M Dp for each p ∈ P . Checking feasibility of break periods, choice of branching candidates and node ordering is done according to sections 5.3.1, 5.3.2, and 5.3.3, respectively. Additional Remarks Obviously, branching on breaks is more efficient than branching on a team’s venue in a specific period if the minimum number of breaks is required, see section 5.4 for details. However, if the problem at hand is not restricted to such HAP sets branching on breaks as proposed in section 5.3 fails. Therefore, we restrict use of the branching scheme in section 6.3.1 to RRTs with the minimum number of breaks and use the branching scheme in section 6.3.1 otherwise. Furthermore, as mentioned in section 5 both branching schemes can not guarantee integer solutions, see table 5.1 for a fractional solution

6.3 Branching Scheme

113

providing no branching candidate according to both concepts proposed above. If no such branching candidate is given we branch as lined out in section 6.3.1 which is sufficient to guarantee integer solutions. Set Partitioning Based Branching The set partitioning problem is a well known integer problem having many applications in various fields. Hence, solution methods are deeply investigated. This holds in particular for branching strategies for the set partitioning problem. Barnhart et al. [4] give an exhaustive overview and emphasize a method developed by Ryan and Foster [75]. This strategy has been successfully applied to B&P frameworks, e.g. in Mehrotra and Trick [60] and Vance et al. [83]. The rule prescribes to find two columns c1 and c2 having fractional variable values in the current optimal solution. Second, two rows r1 and r2 must be determined such that c1 covers r1 and r2 and c2 covers exactly one of them. As shown in Barnhart et al. [4] c1 , c2 , r1 , r2 always exist in a standard set partitioning problem if the current solution is not integral. Then, two subproblems are created. In the first one r1 and r2 are forced to be covered by the same column. In the second subproblem r1 and r2 are forced to be covered by different columns. Obviously, we obtain a partition of solution space. Moreover, the current optimal solution is cut. Projecting this idea to our problem we obtain two different branching techniques outlined below. Pairings in Same/Different MD If r1 and r2 both are of type (6.4) they correspond to two pairings. Hence, branching candidates are pairings r1 =(i 0 1 , j1 ) and r2 =(i 0 2 , j2 ) being contained in a scheduled MD having a fractional variable value if there is another scheduled MD having a fractional variable value, as well, and containing exactly one of these pairings. Then, the first subproblem s1 is restricted to MDs either containing both pairings or none of them. Obviously, this results into an optimal solution to s1 having both pairings covered by the same column. The second subproblem s2 allows no MDs containing more than one of both pairings. Therefore, the optimal solution to s2 has r1 and r2 covered by different columns. In order to consider those restrictions in the column generation process we add constraints (6.39) or (6.40), respectively, to pricing problem M Dp . Constraint (6.39) forces pairing (i1 , j1 ) to be contained in a MD if and only if pairing (i2 , j2 ) is contained as well. It is incorporated in

114


Model 6.8: CG Subproblem – M DpSD xi1 ,j1 − xi2 ,j2

=

0

(6.39)

xi1 ,j1 + xi2 ,j2

≤

1

(6.40)

each pricing problem M Dp , p ∈ P , of s1 . Constraint (6.40) prevents both pairings from being contained in the same MDs and, therefore, is the involved in each pricing problem M Dp , p ∈ P , of s2 . If more than one branching candidate is given we have to decide which candidate to branch on. We evaluated two strategies. The first is to branch on the first candidate which is found and cancel search for further candidates. This strategy is justified by the goal to reduce runtime requirements. The second strategy is to choose among all candidates the one having the highest variable value corresponding to the column covering both rows, r1 and r2 . Here, we consider nearly integer variable values to indicate branching candidates providing low cost solutions. Our tests show a clear superiority of the second strategy and, therefore, everything below is restricted to it. Note, that no cost oriented selection of branching candidates is possible since no specific match (i, j, p) is fixed or forbidden. Fix/Forbid Pairing in Period If r1 is of type (6.4) and r2 is of type (6.5) pairing (i, j)=r 0 1 is contained in at least one MD m with p(m)=r 0 2 as well as in at least one MD m′ with p(m′ ) = p(m) both having fractional variable values. In subproblem s1 both rows are forced to be covered by the same column which means forcing i and j to compete in p. In subproblem s2 both rows are forced to be covered by different columns which means forbidding a match between i and j in p.

Model 6.9: CG Subproblem – M DpF F xi,j + xj,i

=

1

(6.41)

xi,j + xj,i

=

0

(6.42)

In subproblem s1 pairing (i, j) is fixed to be carried out in period p by incorporating constraint (6.41) in pricing problem M Dp and adding


115

constraint (6.42) to each pricing problem M Dp′ , p′ = p. Subproblem s2 is restricted to solutions having pairing (i, j) carried out in period p′ = p. Therefore, pricing problem M Dp employs constraint (6.42). Each MD m with p(m) = p might contain or might not contain pairing (i, j). Therefore, no constraint is added to M Dp′ , p′ = p. Furthermore, we tested another variant employing the fixation of pairings to periods. After detecting a candidate for branching as described above we create |P | child nodes by fixing the pairing to each period. Obviously, the search tree grows wider and less deep by branching that way. However, this method shows no advantage in comparison to the previous method. Again, we have to decide how to select a branching candidate if more than one is given. First, we tested the random strategy to select the first candidate which is found by sequential search. Second, in analogy to a variant outlined in subsection 6.3.1, we select the candidate having the highest variable’s value. Third, we select the candidate leading to the lowest possible cost of the pairing (i, j) to be fixed in period p which is min{ci,j,p , cj,i,p }. Our tests proofed the third approach to be the most efficient one. Note that this approach incorporates a cost orientation which was not possible for Same/Different MD. Additional Remarks We emphasize that both types of branchings are needed to assure integrality for CG Master – SRRT even if no extensions are incorporated. In order to illustrate this we give two infeasible (i.e. fractional) solutions with n = 4 in tables 6.2 and 6.3, respectively. Table 6.2. Fractional Solution Without Candidate Same/Differ Period

p1

p2

p3

MDs

m1 1-2 3-4

m2 1-3 2-4

m3 2-1 4-3

m4 1-3 4-2

m5 1-4 2-3

Variable

0.5

0.5

0.5

0.5

1

The second line of table 6.2 and 6.3, respectively, names MDs which are defined in lines three and four by two matches each. The first lines gives the periods the MDs are assigned to. Variables’ values are given in the last lines. Both solutions are not feasible as can be seen from

116

6 Branch–and–Price Table 6.3. Fractional Solution Without Candidate Fix/Forbid Period

p1

p2

p3

MDs

m1 1-2 3-4

m2 2-1 4-3

m3 1-3 2-4

m4 1-3 4-2

m5 1-4 2-3

Variable

0.5

0.5

0.5

0.5

1

the fractional variables’ values corresponding to the first four MDs. The solutions given in tables 6.2 and 6.3 do not provide any branching candidates according to the first concept and the second concept, respectively, as described above. However, we can find branching candidates according to the second concept and the first concept for solutions given in tables 6.2 and 6.3, respectively. If extensions for CG Master – SRRT are considered there might be optimal fractional solutions not providing a branching candidate. Hence, integer solutions are not necessarily provided so far. Then, we branch according to section 6.3.1. Clearly, if each pairing is fixed to a period and, furthermore, each team’s venue in each period is fixed the whole tournament is fixed. Therefore, including both types of branching steps guarantees integer solutions. Since we have to check a node’s optimal solution for branching candidates according to both approaches we must decide which one to check first. Consequently, both orders were tested. Considering candidates according to subsection 6.3.1 proofed to be more effective. Probably, the reason is that these candidates enable us to select one of them based on cost orientation. In fact, fewer nodes of the search tree are explored. Branching by fixing a team’s venue is employed only if the first two approaches do not provide a candidate. Finally, we discuss the decision which subproblem to explore first. In order to lead the approach towards integer solutions fast we first solve the subproblem forcing two rows to be covered by the same column since it represents a tighter restriction of solution space. Again, we tested other strategies but none could compete with the described one. Summary Considering each branching strategy proposed in sections 6.3.1 and 6.3.1 in preliminary tests we restrict ourselves to the following branching strategies depending on the structure of the problem at hand.


117

If the minimum number of breaks is required we first branch on teams’ breaks according to section 6.3.1. If no such branching candidate is given in the optimal solution being fractional we branch by fixing pairings on periods according to section 6.3.1. If no minimum number of breaks is required we first branch by fixing pairings on periods according to section 6.3.1. If no such branching candidate is given in the optimal solution being fractional we branch by forcing pairings to be contained in the same or in different scheduled MDs, respectively, according to section 6.3.1. If, again, no such branching candidate is given in the optimal solution being fractional we branch by fixing the venue of teams in each period according to section 6.3.1. 6.3.2 Node Order Strategy While working off all nodes of the branching tree we have to decide which node to explore next in each iteration. We propose several strategies below. Common Strategies We employ two well known node order strategies. Depth first search simply prescribes to explore the node first which has been created last. This leads to a single path from the root node to the current node. Hence, the number of created but yet unexplored nodes is rather small and, therefore, memory requirements are low. However, the overall number of explored nodes is rather large unless efficient cost oriented branching decisions are at hand. Breadth first search prescribes to explore the node providing the best (i.e. lowest) lower bound is examined next. Obviously, this leads to a minimum number of explored nodes as long as no information about the optimal solution’s structure and value, respectively, is given. No node having a lower bound lower than the optimal solution’s value gets explored. In fact, we experience lowest runtime requirements to find optimal solutions for rather small problem instances on average. Unfortunately, problems occur for larger instances. Since the search tree grows rather large and many paths are left unpruned we suffer from lack of memory. As a result probability for successful application of the approach gets lower the larger n is. Furthermore, the probability depends on the set of restrictions under consideration. Obviously, finding feasible solutions gets harder if more restrictions are considered. Hence, subtrees can be cut less effectively.

118


Heuristic and Exact Beam Search According to the observations from section 6.3.2 we use beam search as an intermediate approach. Beam search originates from the artificial intelligence community, see Lowerre [57] and Rubin [72] for example. The basic idea is to explore only a fixed number of nodes at each level of the search tree. This number is called beam width w. We select those w nodes having the lowest lower bounds for further exploration. All reamining nodes on the same level of the search tree are discarded which makes the approach a heuristic one. The larger w is chosen the more nodes are explored and, therefore, solution quality and runtime as well as memory requirements rise. If we set w = 1 just a single path from the root node is explored. This either leads to finding a single feasible solution or finishing the procedure without any feasible solution if the path leads to a leaf corresponding to an infeasible problem. Between those extremes, beam search enables us to customize it depending on problem size and constraints taken into account in order to find a promising tradeoff between solution quality and memory requirements. Obviously, the heuristic nature of beam search is a major drawback since we aim at the minimum cost solution. By slight variation we can turn the heuristic beam search into an exact method. Instead of discarding those nodes not selected we preserve them for a backtracking mechanism. If each node is pruned at a specific level k the approach backtracks. Then, among the remaining nodes on level k −1 the w most promising are selected and a new beam is established. Therefore, we have some kind of depth first search order for beams. We experience counterrotating effects when changing w here, too. The larger w is chosen the lower is the number of beams needed to reach the optimal solution as well as the overall run time of our algorithm. On the other hand the smaller w is chosen the lower are memory requirements. Most Binary First Search The number of variables with fractional values in an optimal solution to a LP problem can be considered as a measure for infeasibility. Therefore, the number of variables of a node’s optimal solution having binary values might serve as an indicator for probability to find an integer solution in the corresponding subtree fast. The basic idea is employed in section 5.3.3, as well. In order to find feasible solutions fast we propose to explore the node having the largest number of binary value variables of the current solution.

6.4 Column Generation

119

Summary We evaluate all concepts introduced above. In order to conclude this subsection we shortly line out results. The approaches’ ranking according to run time requirements clearly shows breadth first search superior to beam search being superior to depth first search. According to memory requirements the approaches are ranked in exactly the reverse order. Consequently, we choose recovering beam search for further evaluations since it enables us to benefit from breadth first search ideas just as much as memory capacity allows. Setting beam width w = 15 proofed to be a good choice.

6.4 Column Generation Our CG approach follows exactly the basic idea developed in Gilmore and Gomory [45]: we create a restricted master problem and, alternately, solve it to optimality, determine the minimum reduced cost column (scheduled MD) and insert it into the restricted master problem until there are no columns having negative reduced cost. 6.4.1 Pricing The method employed for finding columns having negative reduced cost highly affects run times. We distinguish between exact methods and heuristics and observe two counterrotating effects: either the number of iterations is small while the run time per iteration is rather large or the number of iterations is large but time per iteration is small. Employing Cplex Pricing problem M Dp , p ∈ P , possibly incorporating restrictions of M Dpext and branching constraints, respectively, can be solved to optimality using Cplex. Obviously, we can solve the pricing problem heuristically by limiting the run time and obtaining the best solution found when time is up. This might result into a scheduled MD having positive reduced cost or no solution at all even if there are solutions having negative reduced cost. Consequently, the CG approach can not guarantee an optimal solution to the master problem. Another method to save run time is to abort the solution process if an arbitrary scheduled MD having negative reduced cost is found. This method guarantees optimality of our CG process since a solution

120


with negative reduced cost is found as long as one exists. If there is no such solution the master problem’s optimal solution is found. However, evaluating these strategies we observe that finding the minimum reduced cost scheduled MD is most effective. That is why we restrict the remainder to it. Polynomial Matching Algorithms Problem M Dp can be represented as a complete graph Kn where nodes correspond to teams and edges correspond to matches between incident teams. Given cost / ci,j,p for each match the problem of finding the minimum reduced cost scheduled MD is reducible to the problem of finding the minimum weight perfect matching of Kn . Edge (i, j), i < j, having weight / ci,j represents the cheaper of both possible matches between teams i and j in p. If both matches have identical cost edge (i, j), i < j, represents a match at i’s home. Lots of papers have been published on this problem mostly using and extending the basic work of Edmonds [35], for example Derigs [27] and Cook and Rohe [17]. The algorithms provided in these articles are well known to run in polynomial time. However, we can further reduce the problem. Since the underlying graph is complete and has an even number of nodes we can represent problem M Dp as maximum weight matching problem. We construct G′ := Kn with weight / c′i,j = ci,j − min{/ / ci,j | i, j, i < j} + 1 for each edge (i, j), i < j. Theorem 6.2. A maximum weight matching of G′ is perfect.

Proof. Let M ′ be a maximum weight matching of G′ . If M ′ is not perfect there are at least two nodes i, j being unmatched. Then, M ′ ∪ {(i, j)} is a matching having higher weight and, hence, M ′ has no maximum weight. ⊓ ⊔

Mehlhorn and Sch¨ afer [59] present an implementation to find minimum weight perfect matchings as well as maximum weight matchings in general graphs. Unfortunately, several requirements introduced in chapter 3 can not be considered when using matching algorithms. This includes region’s capacities and highly attended matches: restrictions (6.36) and (6.37) as well as constraint (6.38) can not be reduced to the minimum weight perfect matching problem. However, most of additional requirements can be considered. Stadium availability and forbidden matches in general can be represented by modifying the corresponding costs: if match (i, j, p) is not possible


121

we set / ci,j,p = Mvio . If both matches between teams i and j are forbidden we simply delete edge (i, j) from the graph corresponding to period p. Obviously, (i, j) can not be chosen for perfect matchings anymore and, hence, no scheduled MD containing a match between teams i and j is constructed for p. Consideration of breaks, opponents’ strengths, and teams’ preferences do not affect the pricing problem’s structure at all. All of them are employed in the master problem and matches cost can be charged with corresponding dual variables. Hungarian Method If a HAP set is given we can further reduce the pricing problem to the minimum weight bipartite matching problem. For each period p there is a partition of teams into teams playing at home and teams playing away. Obviously, matches in p can only be arranged between a team playing at home in p and a team playing away in p. Hence, given a HAP set problem M Dp can be represented by a complete bipartite graph Km,n−m with m ∈ {0, . . . , n}. Note that feasibility of the HAP set requires that m = n2 holds. The minimum weight bipartite matching problem can be solved using hungarian method developed by Kuhn [52]. Additional restrictions can be considered or must be dropped, respectively, in exactly the same way as outlined for the perfect matching algorithm above. Note that a HAP set can be given in advance or can be constructed by the branching scheme in section 6.3.1. Therefore, hungarian method can be employed when solving a HAP-set-based single RRT problem (see section 2.4.2) or when solving a RRT problems requiring a minimum number of breaks. Local Improvement In order to save run times for finding columns having negative reduced cost we propose a local improvement method hereafter. Each column in the base according to the optimal solution to the current restricted master problem serves as a starting point for local search. Since these columns have reduced cost equal to 0 a neighborhood move implying a cost reduction produces a column with negative reduced cost. First, we define neighborhood N (m) of a scheduled MD m: N (m) is the set of scheduled MDs which can be obtained from m 2 by exchanging two teams playing at home. Note that |N (m)| = n8 − n4 for each m ∈ P M . Evaluation of m′ ∈ N (m) can be done implicitly by evaluating the neighborhood move from m to m′ . Let (i1 , j1 ) and (i2 , j2 ) be the

122


matches of m whose teams playing at home are to be exchanged. Then, the move’s value is given by / ci1 ,j2 ,p + / ci2 ,j1 ,p − / ci1 ,j1 ,p − / ci2 ,j2 ,p . Consequently, we sequentially search for neighborhood moves having the lowest value in order to find a pleasant column. Obviously, we can not guarantee to find a column having negative reduced cost by local improvement if there exists any. We can easily incorporate all restrictions known from chapter 3. We define a neighborhood move infeasible if it arranges a forbidden match, if it exceeds the allowed number of highly attended matches, or if it exceeds a region’s capacity. Consequently, we only consider feasible neighborhood moves. Consideration of breaks, opponents’ strengths, and teams’ preferences is possible in analogy to section 6.4.1. Summary We implement the hungarian method designed as an executable as shown in Munkres [67]. For tested problem sizes hungarian method proofed to need almost the same amount of time as Cplex does. We refuse to evaluate algorithms for the minimum cost perfect matching problem and maximum weighted matching problem since solution times are not expected to undershoot those of hungarian method. This is due to the fact that hungarian method uses the structure of a special case of the minimum cost perfect matching problem. Besides, the minimum cost perfect matching problem does not allow to incorporate more restrictions as hungarian method does. The local improvement method does provide columns having negative reduced cost faster than Cplex does. If no column is found by local improvement we employ Cplex afterwards in order to guarantee optimality. Despite of the lower run time requirements to find new columns the column generation process converges significantly slower since Cplex provides columns having minimum reduced costs. The overall time for the column generation process is less if Cplex is used. Analogously, obtaining the optimal solution from Cplex is more efficient than obtaining the first column having reduced cost. Due to these results and the fact that Cplex can handle each type of additional constraints we restrict the remainder to pricing problems being solved by Cplex to optimality. 6.4.2 Column Management Column management covers three activities: generating initial columns in order to construct the first restricted master problem, generating


123

profitable columns during the column generation approach, and deleting columns not profitable anymore in order to reduce the restricted master problem’s size. Generating Initial Columns Initial columns for the restricted master problem have to provide at least one feasible solution to the problem represented by the current node. The constraints in the current node are composed of constraints of the original problem and of constraints resulting from branching steps. Each constraint of the restricted master problem originates from the original problem. We construct feasible solutions according to constraints in CG Master – SRRT by using a generation scheme originating from Kirkman [50], namely the polygon technique. If a minimum number of breaks is required we generate feasible solutions by the generation scheme given in Schreuder [76]. In order to avoid breaks in the second period (which is mentioned to be a common real world requirement in chapter 3) we assign MDs to periods in reversed order. Note that both generation schemes can not guarantee feasible solutions according to changing opponents’ strengths. Therefore, variable esvio is introduced and is used in terms (6.15), (6.16) and (6.3a). This results into formal feasibility of solutions not obeying the changing opponents’ strengths requirement. Due to the high cost Mvio associated with esvio the optimization method will lead to esvio equaling 0 and, therefore, SLSs fulfilling changing opponents’ strengths requirements. The constraints of M Dp , p ∈ P , are fulfilled by each RRT and, hence, by both generation schemes. Constraints of M Dpext , p ∈ P , on the one hand and constraints derived from branching decisions on the other hand can not be considered by both generation schemes. Scheduled MDs not obeying those constraints are called infeasible MDs in the remainder. We incorporate infeasible MDs constructed by the generation schemes as initial columns and, consequently, have to assure that these columns are banned during the column generation process. Therefore, we set cost cm to Mvio if scheduled MD m is infeasible. Instead of generating a single scheduled MD per period by using the generation schemes we propose to construct |P | scheduled MDs per period. The generation schemes yield |P | different MDs forming a single RRT. We assign each of them to each period and obtain |P |2 different scheduled MDs. The underlying idea is to provide a solution as good as possible by the initial restricted master problem.

124


Alternatively, we suggest to employ columns corresponding to the father node’s optimal solution as initial columns. When doing this we have to check whether a MD m is feasible according to constraints of the father node but gets infeasible by the constraint derived from the branching step. If so we set cm = Mvio in the current node. This methodology is justified by the fact that the father node and the current node differ in only one branching step. Therefore, the father node’s optimal solution might just slightly differ from the optimal solution or at least a good solution of the current node. Obviously, we can not use this idea for the root node. We tested all variants outlined above. The following strategy proofed to be the most efficient one (measured in run times) and is the one we restrict all what follows to. • For the root node we construct |P |2 initial columns by assigning each MD obtained from the generation scheme of Schreuder [76] to each period. • For each other node we pass columns contained in the father node’s optimal solution as well as those from the generation scheme as initial columns to the child nodes. Generating Columns Since we have |P | decomposed problems M Dp , p ∈ P , resulting from the pricing problem M D there are several decisions to be made about the solution process. In order to find the overall minimum reduced cost scheduled MD we have to solve all single decomposed problems iteratively. Hence, we determine |P | scheduled MDs which all might have negative reduced cost. Consequently, we propose to insert all scheduled MDs with negative reduced cost into the restricted master problem. Alternatively, we can solve decomposed problems until we have found the first scheduled MD with negative reduced cost. Therefore, we solve the decomposed problems in decreasing order of dual variable βp . βp is a constant element of the reduced cost of MDs assigned to period p and, therefore, the problem M Dp having highest βp is most likely to produce a scheduled MD having negative reduced cost. Testing both variants made clear that solving each single decomposed problem is more efficient.


125

Discarding Columns During the CG process lots of scheduled MDs are generated and are inserted into the restricted master problem. Nearly all of them turn useless later on when they are not part of the current optimal solution’s base anymore. In order to reduce solution times for the restricted master problem it is substantial to prevent it from growing too large. There are two popular ideas to choose columns to be deleted from the restricted master problem: • A column is deleted whenever it has not been part of the optimal base for a given number of iterations. • A column is deleted whenever its reduced cost are larger than a given threshold. According to preliminary tests the best choice for our problem is to delete a scheduled MD when it has not been part of the base for 5 iterations. 6.4.3 Lower Bounds One major drawback of column generation is that optimality of solutions is often proven much later than the solution is found (also called tailing-off effect). Hence, many iterations are spent useless (in terms of finding the master’s optimal solutions). Consequently, we propose a lower bound of the current node’s optimal solution value. This lower bound is calculated according to the current restricted master’s optimal solution and the optimal solutions to the pricing problems M Dp . If the lower bound is larger than the best feasible solution’s value so far we cancel the column generation process. Then, the subtree corresponding to the current node is pruned. With respect to the B&P scheme we consider the set P Mkf ⊆ P M of scheduled MDs being feasible according to the current node k. Note that each branching step is represented by additional constraints of M Dp and, therefore, can be fully represented by reduction of P M . Furthermore, we have to take into account the set P Mkinf of scheduled MDs introduced into the restricted master as initial columns but being infeasible according to the current node k. Let P Mk′ = P Mkf ∪ P Mkinf be the set of MDs under consideration in node k. Lower Bound based on Minimum Reduced Cost Lasdon [54] proposes a lower bound for the optimal solution’s value of a master problem having a set partitioning structure. We pick this idea up

126


and adapt it to all variants of our master problem under consideration. We denote the reduced cost of scheduled MD m by cm hereafter. Lemma 6.1. For each p ∈ P equation minm∈P Mk′ ,p(m)=p {cm } = min minm∈P M f ,p(m)=p cm , 0 holds. k

Proof. Note that cm ≥ 0 holds for each m ∈ P Mkinf due to . Obvicm = Mvio and cm ∈ R holds for each m ∈ P Mkf

ously, minm∈P Mk′ ,p(m)=p {cm } = min minm∈P M f ,p(m)=p cm , 0 holds k if minm∈P M f ,p(m)=p {cm } ≤ 0. However, minm∈P M f ,p(m)=p {cm } > 0 k k might hold if solely infeasible MDs form period p of the current restricted master’s solution. If so, minm∈P M inf ,p(m)=p {cm } = 0 holds k since at least one infeasible MD is part of the current solution. ⊓ ⊔ Theorem 6.3. Suppose an optimal solution to the current restricted master problem having value zcur is given. Then, a lower bound of the optimal solution’s value to the master problem (considering breaks and changing opponents’ strengths) is given by

zcur +

min

p∈P

min

m∈P Mf ,p(m)=p

{cm } , 0 +

(γi,p + δi,p − ǫ) bri,p +

i∈T p∈P ≥2

esci,p

i∈T p∈P ≥2

ζi,S ′ ,p −

S ′ ∈S

i∈T



ηi 

p∈P ≥2



esci,p − esvio  .

Proof. Subtracting the right hand side of (6.5) and the left hand side of (6.5), respectively, multiplied with corresponding dual variables βp from objective function (6.3) leads to equation (6.43).

m∈P Mk′

cm ym −

p∈P

βp =

m∈P Mk′

cm ym −

p∈P

βp

ym (6.43)

m∈P Mk′ ,p(m)=p

Doing so for (6.4), (6.9), (6.10), (6.11), (6.13), and (6.14), as well, leads to (6.44). Note that all dual variables used in this second step are not positive.


cm ym −

m∈P Mk′

i∈T

cm ym −

m∈P Mk′

αi,j

i∈T p∈P ≥2

ǫ



i∈T p∈P

i∈T S ′ ∈S p∈P ≥2

i∈T

ηi 

p∈P ≥2

ηi ≥

i∈T

p∈P ≥2

ym −

ym −

→ j∈T,j=i m∈P M ′ ,(− i,j)∈m,p−1≤p(m)≤p k

→ j∈T,j=i m∈P M ′ ,(− j,i)∈m,p−1≤p(m)≤p k



bri,p  −



 δi,p 

ζi,S ′ ,p − escmax

m∈P Mk′ ,p(m)=p

 γi,p  

i∈T p∈P ≥2



βp

127

γi,p −

m∈P Mk′ ,(i,j)∈m

i∈T j∈T,j>i

S ′ ∈S

i∈T p∈P ≥2

i∈T

p∈P

αi,j −

i∈T j∈T,j>i

δi,p − (n − 2) ǫ −

p∈P ≥2

βp −

p∈P



ζi,S ′ ,p 



 ym − bri,p  − (6.44)  ym − bri,p  −

j∈S ′ ,j=i m∈P Mk′ ,(i,j)∈m,p−1≤p(m)≤p



esci,p − esvio 





ym − esci,p  −

The terms being subtracted from m∈P M cm ym on the left hand side of (6.44) sum up to the dual solution’s value and, therefore, to the current optimal solution’s value of the restricted master due to duality theory. Furthermore, we extract scheduled MDs’ reduced cost on the right hand side and obtain (6.45).

128


cm ym − zcur ≥

m∈P Mk′

m∈P Mk′



ym cm −



αi,j − βp(m) −

(i,j)∈m,j>i



  γi,p(m) + δj,p(m) +γi,p(m)+1 + δj,p(m)+1  − + ,.+ ,.

− → (i,j)∈m

if p(m)≥2



if p(m)≤|P |−1



  ζi,S(j),p + ζj,S(i),p +ζi,S(j),p+1 + ζj,S(i),p+1  + + ,.+ ,.

− → (i,j)∈m

if p(m)≥2

(6.45)

if p(m)≤|P |−1


i∈T p∈P ≥2

ζi,S ′ ,p esci,p −

i∈T S ′ ∈S p∈P ≥2

i∈T



ηi 

p∈P ≥2




We identify reduced cost of scheduled MD m ∈ P Mk′ (see (6.17), (6.18), and (6.19)) in lines 2, 3, and 4 of (6.45). With m∈P Mk′ ,p(m)=p ym = 1 for each p ∈ P and minm′ ∈P Mk′ ,p(m′ )=p {cm′ } ≤ cm for all m ∈ P Mk′ with p(m) = p we transform inequality (6.45) into (6.46).

cm ym ≥ zcur +

m∈P Mk′

p∈P

min

m∈P Mk′ ,p(m)=p


i∈T p∈P ≥2

i∈T p∈P ≥2

{cm }+

esci,p

S ′ ∈S

ζi,S ′ ,p −

i∈T

(6.46) 

ηi 

p∈P ≥2




Employing lemma 6.1 in (6.46) we obtain (6.47).


cm ym ≥ zcur +

m∈P Mk′

min

p∈P

i∈T

cm , 0 +


p∈P ≥2

min

m∈P Mkf ,p(m)=p

esci,p

i∈T p∈P ≥2

ζi,S ′ ,p −

S ′ ∈S

i∈T

129

1

(6.47) 

ηi 

p∈P ≥2

Term (6.47) directly implies theorem 6.3.




⊓ ⊔

Our proof is restricted to the case where constraint (6.5) is formulated as equation. Adaption to constraint (6.5) being “no more than” constraint is straightforward. First, equation (6.43) is replaced by (6.43’) since βp ≤ 0 ∀p ∈ P .

m∈P Mk′

cm ym −

p∈P

βp ≥

m∈P Mk′

cm ym −

p∈P

βp

ym (6.43’)

m∈P Mk′ ,p(m)=p

Then, relation (6.45) is obtained analogously and can be transformed to (6.46) since cm ≤ 0 for each m ∈ P Mk′ . Hence, the lower bound stated in theorem 6.3 holds no matter which formulation of the master problem is chosen. Therefore, we can obtain lower bounds while solving the master problem with “no more than” constraints even if this formulation does not provide an optimal solution to the master problem (see section 6.2.1 for details). Another restriction of the proof is that minimum number of breaks and changing opponents’ strengths are considered. Note that, again, adaption to other cases is simple and, consequently, we can state four lower bounds depending on which of both types of constraints are considered, see table 6.4. Note, that definition of cm depends on the structure of the master’s problem as well (see sections 6.2.1 and 6.2.2 for details). A fairly convenient property of this lower bound is that nearly no additional computational effort is to be made: Dual variables are known from the current optimal solution to the restricted master problem and the minimum reduced cost of scheduled MDs’ is computed in order to find pleasant columns for each period, anyway. Moreover, this lower bound implies a customization to the current node k of the search tree. Branching constraints are incorporated in the matching subproblem restricting solution space of M Dp .

130


Table 6.4. Lower bounds for CG master problem according to reduced cost additional constraints

lower bound

zcur + p∈P min minm∈P M f ,p(m)=p cm , 0 k min no of breaks zcur + p∈P min minm∈P M f ,p(m)=p cm , 0 + k ≥2 (γi,p + δi,p − ǫ) bri,p i∈T p∈P opponents’ strengths zcur + p∈P min minm∈P M f ,p(m)=p cm , 0 + k c S ′ ∈S ζi,S ′ ,p!− i∈T p∈P ≥2 esi,p c ≥2 esi,p − esvio i∈T ηi p∈P min no of breaks, zcur + p∈P min minm∈P M f ,p(m)=p cm , 0 + k (γ + δ − ǫ) bri,p + opponents’ strengths i∈T p∈P ≥2 i,pc i,p S ′ ∈S ζi,S ′ ,p!− i∈T p∈P ≥2 esi,p c i∈T ηi p∈P ≥2 esi,p − esvio none

Lower Bound Based on Feasible Dual Solution Farley [37] proposes a lower bound for the optimal solution’s value to a general master problem. Its computation requires a minimum cost MD problem having a non-linear objective function to be solved in our case. Therefore, with respect to the computational effort we refuse to employ this method. Instead, we develop a lower bound based on the same idea. Our lower bound can be computed via solving a linear minimum cost MD problem (lined out to be solvable in polynomial time for most variants in section 6.4.1) at the cost of being less tight than the one proposed in Farley [37]. First, we give the dual problem CGD to the LP relaxation of the master’s formulation considering a minimum number of breaks and changing opponents’ strengths. The dual problem to the restricted master problem differs from CGD by (6.49) not being stated for each m ∈ P Mk′ but for a subset P Mk′′ ⊂ P Mk′ of scheduled MDs currently under consideration. Theorem 6.4. Suppose an optimal solution to the current restricted master value zcur is given. Let minproblem having {cm } f m∈P M ,p(m)=p k for each p ∈ P . If lbp = max max {−cm +cm } , 1 f m∈P M ,p(m)=p k maxm∈P M f ,p(m)=p {−cm + cm } < 0 and − maxp∈P {lbp } · i∈T ηi ≤ k

6.4 Column Generation Model 6.10: CGD max

αi,j +

i,j∈T,j>i

βp +

p∈P

(γi,p + δi,p ) + ǫ +

i∈T p∈P ≥2

ζi,S ′ ,p + escmax

i∈T S ′ ∈S p∈P ≥2

s.t.

131

ηi

(6.48)

i∈T

αi,j + βp(m) +

i,j∈T,i<j,(i,j)∈m

γi,p + γi,p+1 + δi,p + δi,p+1 + ζx ≤ cm ∀ m ∈ P M

(6.49) ≥2

(6.50) (6.51)

− γi,p − δi,p + ǫ − ζi,S ′ ,p + ηi

≤0

∀ i ∈ T, p ∈ P

≤0

∀ i ∈ T, p ∈ P ≥2

−

≤ Mvio

S ′ ∈S

ηi

(6.52)

i∈T

αi,j , γi,p , δi,p , ǫ, ζi,S ′ ,p , ηi

≤0

∀ i, j ∈ T, i < j, p ∈ P ≥2 , S ′ ∈ S (6.53)

βp

∈R

∀p∈P

(6.54)

Mvio hold, then, a lower bound of the optimal solution’s value to the master problem (considering breaks and changing opponents’ strengths) is given by zcur · maxp∈P {lbp }. Proof. Let os be the optimal solution to the restricted master’s dual. The optimal solution’s value to the restricted master problem equals the value of os due to duality theory. By adding columns during the CG process the current dual solution might get infeasible. This is formally indicated by a scheduled MD m ∈ P Mkf \ P Mk′′ where (6.49) is violated according to os. If each m ∈ P Mkf implies a negative left hand side of (6.49), i.e. maxm∈P M f ,p(m)=p {−cm + cm } < 0, we construct a dual solution os′ k

fulfilling (6.49) for each m ∈ P Mk′ by setting os′ = os · maxp∈P {lbp } (here, multiplying a solution by a scalar means multiplying each varimin

able by it). Note that

{cm } f m∈P M ,p(m)=p k max {−cm +cm } f m∈P M ,p(m)=p k

< 1 might hold due to

the fact that solely infeasible MDs might be assigned to period p. If so, there is at least one scheduled MD in P Mkinf fulfilling (6.49) with equality and, hence, lbp can not be chosen lower than 1.

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Multiplying ǫ, γi,p , and δi,p by maxp∈P {lbp } leads to restriction (6.50) being fulfilled since maxp∈P {lbp } ≥ 1. The same is true for ζi,S ′ ,p and ηi multiplied by maxp∈P {lbp } fulfilling constraint (6.51). If, additionally, − maxp∈P lbp i∈T ηi ≤ Mvio then (6.52) holds ′ ′ for os and, hence, os is feasible to the master problem’s dual problem. The value of os′ equals zcur · maxp∈P {lbp } and, consequently, forms a lower bound for the master problem. ⊓ ⊔ Varying equality (6.5) to “no more than” constraints changes (6.54) to (6.54’) but affects neither the lower bound nor the proof’s idea. βp ≤ 0 ∀ p ∈ P

(6.54’)

In analogy to section 6.4.3 the proof only covers the lower bound for the master problem considering breaks and changing opponents’ strengths, here. Adaption to the master problem not considering both kind of restrictions is straightforward. In each case the lower bound is given as min

zcur · maxp∈P {lbp } with lbp = max

{cm } f m∈P M ,p(m)=p k max {−cm +cm } f m∈P M ,p(m)=p k

,1

if maxm∈P Mf ,p(m)=p {−cm + cm } < 0. Note that the definition of cm depends on the restrictions considered by the original problem. Whether (−cm + cm ) < 0 holds for each m ∈ P Mf can easily be checked by solving a maximum cost MD problem with cost (−/ ci,j,p + ci,j,p ). This has to be done in order to compute lbp , any way. Computing − maxp∈P lbp i∈T ηi in order to make sure that ′ (6.52) is fulfilled by os does not imply additional computational effort either. Hence, checking conditions for the lower bound to be valid can be done easily. Note that minm∈P M f ,p(m)=p {cm } is constant for a k specific node k and independent from the CG process. Therefore, we compute the minimum cost scheduled MD for each period solely once per node. However, computing maxm∈P Mf ,p(m)=p {−cm + cm } doubles the number of matching subproblems to be solved during CG. Again we profit from the branching constraints being employed in the pricing problem so they are considered implicitly. Summary We evaluate the lower bounds proposed above with respect to their influence on run times. We tested the lower bounds for each variant of CG Master – SRRT and additional constraints as well as forbidden matches. Instances are constructed having 8 to 12 teams. Probability

6.5 Upper Bounds

133

for a match to be forbidden was 0.1 if forbidden matches are considered. Reduction of average run times per node are given in table 6.5. Table 6.5. Reduction of run times per node by lower bounds breaks strength Pπ

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ 1.0 0.9 1.0 0.9 1.0 0.9 1.0 0.9

l.b. red.cost (6.4.3) 11% 11% 5% 14% 4% 11% 30% 31% l.b. dual sol. (6.4.3) — — — 13% — — — —

The lower bound based on dual feasible solutions can not lower the average run time per node of the search tree. Run time savings by aborting the CG process before optimality is consumed by additional computational effort in order to compute the lower bound. For the instance class considering changing opponents’ strengths and forbidden matches we observe a severe reduction of run times. We conjecture the reason to be the enormous tailing off effects resulting from the rather large value of esvio if no lower bound is applied. The lower bound based on minimum reduced cost leads to significant lower run times per node in each single problem class. Moreover, it outperforms the lower bound based on dual feasible solutions for the instance class considering changing opponents’ strengths and forbidden matches. Using the lower bound based on minimum reduced cost leads to a significant reduction of run times per node and number of iterations of the CG process. Reduction is not less than 4% over all instances and more than 14% on average. Applying both lower bounds further reduced run times for instances considering changing opponents’ strengths and forbidden matches in comparison to applying only one lower bound. In order to present a unique approach and according to these results we restrict ourselves to applying the lower bound based on reduced cost for each problem class in the remainder.

6.5 Upper Bounds In order to get feasible solutions to the binary master problem we propose a hill-climbing oriented heuristic below. Dinitz and Stinson [28] develop a hill-climbing algorithm for finding one-factorizations of

134


a complete graph being equivalent to a single RRT. While there are special cases of one-factorizations which are easy to find (canonical one-factorizations for example) the algorithm is advisable to us since resulting one-factorizations are not restricted to special classes. The construction of one-factorizations is done using random elements. Even so, the algorithm never gets stuck unable to create a one-factorization in numerous test runs. Therefore, it is reasonable to use this idea in order to complement one-factorizations based on one-factors given as a prerequisite and being randomly constructed (from an design theory point of view). Additionally, run times for creating random one-factorizations are reported to be small. We choose scheduled MDs from the current node’s optimal solution and use them as a precondition for the hill-climbing algorithm of Dinitz and Stinson [28]. Due to the cost minimization master problem scheduled MDs having large variables’ values might serve as an adequate start in order to find a low cost single RRT. Since we intend to complement the single RRT to low cost we replace the random choice in the construction step in Dinitz and Stinson [28] (who do not consider any costs) by a cost oriented one. Additional requirements (such as forbidden matches and stadium availabilities) can be considered by eliminating those construction steps leading to violations. Consequently, probability to find a single RRT is decreased. We define a partial scheduled MD as set of matches where each team contributes no more than once and which is assigned to a period. We define a partial single RRT with n teams as set of n−1 partial scheduled MDs where exactly one partial scheduled MD is assigned to each period and where no match is contained more than once. First, we choose those scheduled MDs from the current node’s optimal solution having a variable’s value exceeding a given threshold y t . If there are more than one of those scheduled MDs assigned to the same period we choose the one having the highest variable’s value. These initial scheduled MDs are fixed as far as the complementing heuristic is concerned. The initial partial single RRT is completed by empty scheduled MDs assigned to those periods where no scheduled MD is chosen from the current node’s solution due to low variable values. After initializing a partial single RRT t the heuristic aims at changing t in each construction step by either adding a match or by exchanging a match contained in t by a match not contained yet. Therefore, we execute two construction steps from Dinitz and Stinson [28] in alternating order.

6.5 Upper Bounds

135

• We consider all teams i and periods p such that i does not contribute to a match contained in the partial scheduled MD mp assigned to p. For each tuple (i, p) consider all teams j where the pairing (i, j) is not contained in t. As shown in Dinitz and Stinson [28] there is at least one triple (i, j, p) if t is not a single RRT. Each triple (i, j, p) corresponds to matches (i, j, p) and (j, i, p). Among all matches corresponding to a triple we choose the one having lowest cost. Let (i′ , j ′ , p′ ) be the chosen match in the following. Obviously, i′ does not contribute to a match in mp′ . We distinguish two cases: – If j ′ does not contribute to mp′ either we add (i′ , j ′ , p′ ) to mp′ and obtain a partial single RRT having one more match than t has. – If j ′ does contribute to mp′ in a match against team i′′ = i′ we delete the match of j ′ against i′′ from mp′ , add (i′ , j ′ , p′ ) to mp′ , and obtain a partial single RRT having the same number of matches as t has. • We consider all pairs (i, j) for each period p such that neither i nor j contribute to M Dp . Among all matches corresponding to such a triple (i, j, p) we choose the one having lowest cost, say (i′ , j ′ , p′ ). We distinguish three cases: – If i′ does not play against j ′ in t we add (i′ , j ′ , p′ ) to mp′ and obtain a partial single RRT having one more match than t. – If i′ plays against j ′ in t in period p and mp is not an initial scheduled (and, therefore, fixed) MD we delete the match of i′ against j ′ in p, add (i′ , j ′ , p′ ), and obtain a partial single RRT having the same number of matches as t has. – If i′ plays against j ′ in t in period p and mp is an initial scheduled MD from the CG master problem we consider inserting (i′ , j ′ , p′ ) infeasible. Therefore, we search all triples (i, j, p) in ascending order of corresponding matches’ cost in order to find a feasible match. If there is no feasible introduction we skip the construction step. Furthermore, forbidden matches introduced in chapter 3 can be easily incorporated by considering the insertion of a match infeasible if the match is infeasible. Stadiums being not available can be considered a special case of forbidden matches. Therefore, we can take care of stadium availability within the upper bound procedure. If a minimum number of breaks is to be reached and the branching strategy according to section 6.3.1 is employed we use the (partial) HAP set generated by branching steps in order to guide the upper bound

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heuristic. A triple (i, j, p) and the corresponding matches, respectively, are considered infeasible if i and j have the same venue in p. Otherwise, the HAP set implicates the feasible match corresponding to (i, j, p). Although consideration of forbidden matches, stadium availability, and HAP sets is straightforward it severely reduces probability of finding a feasible solution. Therefore, we abort a heuristic’s run when no feasible solution is found after a given number of iterations. In order to evaluate the upper bound heuristic we determine several key figures. We investigate problem instances having the basic structure, considering forbidden matches (Pπ = 0.8), considering stadium availability (PπS = 0.8), and requesting the minimum number of breaks with up to 14 teams. We give the fraction of the time necessary to find the first feasible solution (f.s.), to find the optimal solution (o.s.), and proofing optimality (opt.) employing the heuristic and not employing it, respectively. Furthermore, we give the fraction of heuristic’s runs leading to a feasible solution (suc.). We set y t = 0.5 and the maximum number of iterations to 200. Results are given in table 6.6. Table 6.6. Key figures for upper bound heuristic single RRT problem n f.s. o.s. opt. 6 8 10 12 14

0.60 0.92 0.97 0.42 0.92 0.84 0.05 0.93 0.90 0.01 1.00 1.01 0.00 1.03 1.04

forbidden matches (Pπ = 0.8) n f.s. o.s. opt. suc.

suc. 0.89 0.94 0.96 0.97 0.93

6 8 10 12 14

0.69 1.00 1.00 1.00 0.68 0.99 1.07 0.75 0.11 1.01 1.01 0.76 0.01 0.98 0.98 0.82 0.00 1.00 1.01 0.73

stadium availability (PπS = 0.8) n f.s. o.s. opt. suc. 6 8 10 12 14

0.85 1.12 1.09 0.32 0.96 0.81 0.12 0.88 0.90 0.02 1.02 1.01 0.00 1.00 1.01

0.82 0.88 0.79 0.73 0.63

For instances of the basic problem, problem instances considering forbidden matches, and problem instances considering stadium avail-


137

ability the upper bound heuristic mostly succeeds in finding a feasible solution. The highest fraction of successful runs is reached for the basic problem. Only for one class of instances the fraction is less than 0.73. We clearly observe that the time necessary to find the first feasible solution is substantially decreased by the upper bound heuristic for the basic single RRT problem and instances considering forbidden matches and stadium availability, respectively. For each class of instances a feasible solution is found within 1.3 seconds. However, this does not significantly affect the time necessary to find the optimal solution or proof optimality, respectively. These times are slightly decreased for several classes of small instances but for none having more than 10 teams. Therefore, we refuse to use the upper bound heuristic in the remainder. We used the upper bound heuristics for instances requiring the minimum number of breaks, as well. However, almost no run of the upper bound heuristic leads to a feasible solution unless a full HAP set is given, i.e. the current node’s depth is at least n. This depth is reached too rarely for the upper bound heuristic to be of any relevant use.

6.6 Computational Results In order to check the efficiency of the B&P algorithm introduced in sections 6.1 to 6.5 we pick up the problem instances the computational study in section 3.3.2 is based on. We give results in the following according to each problem structure incorporated in the B&P approach. We line out two characteristics for each class of problem instances. First, we give the quality of the lower bound obtained by the CG model measured as the fraction by which the lower bound according to the CG model is larger than the lower bound provided by the LP relaxation of models given in chapters 2 and 3. In order to provide an intuitive understanding we transform all cost ci,j,p to be non-negative and, therefore, obtain a fraction which can not be less than 0 according to theorem 6.1. Second, we give the run time and compare it to the one determined in section 3.3.2. These characteristics are given in tables 6.7 to 6.14. We observe that no lower bound given by the CG model is more than 2.5% higher than the one given by the LP relaxation in table 6.7. Here, the lower bound is increased by more than 2% only for the class of smallest instances. Run times show the exponential characteristic discovered in section 3.3.2, as well. Moreover, run times are substantially larger than the ones obtained by employing Cplex.

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6 Branch–and–Price Table 6.7. Comp. Results for Single RRT problem (B&P) n l.b. 6 8 10 12 14

r.t. r.t. Cplex

2.49% 0.03 1.56% 0.38 1.50% 7.34 1.27% 105.66 1.91% 3723.6

0.01 0.05 0.37 2.03 34.43

Table 6.8. Comp. Results for Forbidden Matches (B&P) Pπ =0.9 Pπ =0.8 Pπ =0.7 n l.b. r.t. l.b. r.t. l.b. r.t. 6 8 10 12 14

0.86% 0.05 1.59% 0.45 1.34% 6.73 1.42% 79.61 1.15% 6293.13

1.88% 0.03 0.95% 0.29 1.43% 5.28 1.27% 110.83 1.22% 5100.05

1.92% 0.04 1.81% 0.23 1.41% 3.94 1.44% 81.84 1.20% 3419.63

Pπ =0.6 Pπ =0.5 r.t. l.b. r.t. n l.b. 6 8 10 12 14

3.91% 0.03 0.90% 0.35 1.09% 6.46 1.32% 127.67 1.19% 5979.23

1.63% 0.03 1.74% 0.24 1.15% 6.74 1.19% 98.06 1.23% 3872.82

As reported for computational results achieved employing Cplex in section 3.3.2 there is no clear vision of the influence of n or Pπ on run times using the B&P approach given in table 6.8. However, they fit into the order of magnitude stated for the basic problem (Pπ =1) and, hence, are higher than those achieved via Cplex. The same is true for the increase of the lower bound. The increase seems to be smaller if Pπ is smaller by tendency. Again, no increase is larger than 2% for instances having more than 6 teams. In table 6.9, mostly, run times are smaller than those for corresponding problem classes of the basic problem and problems considering forbidden matches. We observe a clear decrease of run times the lower PπS is chosen. This directly corresponds to the observation according to


139

Table 6.9. Comp. Results for Stadium Availability (B&P) PπS =0.9 PπS =0.8 PπS =0.7 n l.b. r.t. l.b. r.t. l.b. r.t. 6 8 10 12 14

2.89% 0.03 0.58% 0.29 1.49% 8.29 1.16% 164.03 1.31% 4657.30

0.19% 0.05 0.91% 0.36 1.04% 4.35 1.54% 140.78 1.28% 2647.32

0.03% 0.04 0.61% 0.17 1.30% 1.24 0.47% 42.67 0.98% 1843.94

solution times using Cplex. The gap between run times of Cplex and the B&P approach widens here. The lower bound’s increase is smaller by tendency than for the basic problem (PπS = 1) and, moreover, is smaller if PπS is chosen smaller by tendency. For an explanation of this effect we refer to results according to the HAP-set-based single RRT problem and comments. However, we suppose a correlation between the decrease of lower bound’s quality and the widened gap between run times of Cplex and our B&P approach. Table 6.10. Comp. Results for HAP–set–based Single RRT Problem (B&P) n l.b. 6 8 10 12 14 16

r.t.

0.00% 0.00 0.00% 0.01 0.00% 0.40 0.00% 3.12 0.00% 76.42 0.00% 3306.65

For HAP-set-based single RRT problems the lower bound is not increased in comparison to the one given by the LP relaxation as stated in table 6.10. This is due to theorem 6.1. Solutions to the LP relaxation being cut by the CG reformulation are those violating at least one odd set constraint (6.8). Given a HAP set there is no violation of (6.8) in solutions to the LP relaxation since a HAP set is a bipartition of teams into teams playing at home and away, respectively, for each period. We can project the thought above to instances considering stadium availability. Two subsets of teams are given for each period. No two

140


teams being contained in the same set can compete in the corresponding period. This reduces the number of possible violations of (6.8). Consider, for example, teams i and j both playing away in period p. ′ ′ Then, i and j can not be contained in an odd set T ⊂ T , |T | = 3 with i′ ∈T ′ j ′ ∈T \T ′ xi′ ,j ′ ,p + xj ′ ,i′ ,p < 1. In comparison to table 6.9 run times are significantly decreased in table 6.10. Note that HAP-set-based single RRT problems can be considered as special cases of problems considering stadium availability with PπS = 0.5. Then, results in table 6.10 correspond to the observation that run times decrease if PπS is lowered. However, the gap between run times obtained by Cplex and the B&P approach, respectively, is wider than for the basic problem due to the observation according to the lower bounds given above. Table 6.11. Comp. Results for Highly Attended Matches (B&P) n Pa amax l.b. 6 8 10 12 14

0.2 0.2 0.2 0.2 0.2

1 1 2 2 2

r.t.

8.24% 0.04 4.70% 0.77 1.92% 6.60 1.96% 143.36 1.96% 7965.26

In table 6.11 problem instances incorporating highly attended matches are considered. We observe a quite large increase of the lower bounds quality for small instances having no more than 8 teams. However, for larger instances quality is on the level observed for other classes of instances above: increase of lower bound is less than 2% in comparison to the lower bound given by the LP relaxation. Again, run times are significantly higher than those achieved with Cplex. Considering regions we obtain a slightly better lower bound quality than for those problem instances taken into account so far. Only one third of classes of instances considered in table 6.12 provide a lower bound’s quality of less than 2%. However, no lower bound’s quality is larger than 4% and, therefore, run times are higher than those achieved using Cplex. Again, we observe the correlation between lower bound’s quality and the gap between running times by employing Cplex and the B&P approach, respectively: the lower bound’s solution is best for instances having four regions and, consequently, the gap is smallest for these instances. Nevertheless, it remains significant.


141

Table 6.12. Comp. Results for Regions (B&P) n |R′ | CR′

|R| = 2 l.b.

6 8 10 12 14

3.51% 0.07 2.93% 0.53 1.70% 4.92 2.53% 132.47 1.70% 10131.70

2/4 3/5 4/6 5/7 6/8

2/3 2/3 3/4 3/4 4/5

n 6 8 10 12 14

|R| = 3 CR′ l.b.

r.t. |R′ |

2/2/2 2/2/2 0.94% 0.05 2/3/3 2/2/2 2.11% 0.43 3/3/4 2/2/3 2.84% 19.35 4/4/4 3/3/3 0.88% 252.60 4/5/5 3/3/3 2.66% 23597.20

|R| = 4 l.b. CR′

|R′ |

r.t.

r.t.

1/1/2/2 1/1/2/2 2.52% 0.07 2/2/2/2 2/2/2/2 1.43% 0.36 2/2/3/3 2/2/2/2 3.50% 10.07 3/3/3/3 2/2/2/2 3.67% 630.25 3/3/4/4 2/2/3/3 2.75% 45538.86

Table 6.13. Comp. Results for Breaks (B&P) n

l.b.

r.t.

6 0.88% 2.55 8 1.40% 249.97 10 0.83% 63485.47

Table 6.13 provides computational results according to the minimum number of breaks (we forbid breaks in the second period, here). Again, the lower bound is rather poor and, therefore, run times for problems with up to 8 teams are higher than those needed by Cplex. Remarkably, we can solve instances with 10 teams to optimality in less than 18 hours on average while Cplex failed. However, we mainly accredit smaller run times to the branching scheme developed in section 5.3. Only the changing opponents’ strengths requirement is implemented for the B&P approach. Here, we obtained worse than average lower bound’s quality as can be seen in table 6.14: no lower bound’s quality exceeds 1.7%. Remarkably, for |S| = 2 no lower bound’s quality is larger than 0.6%. Consequently, run times are substantially larger than those obtained with Cplex.

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Table 6.14. Comp. Results for Changing Opponents’ Strengths (B&P) |S| = 2 |S| = n2 run. t. n l.b. run. t. l.b. 8 0.00% 0.45 1.24% 2.19 10 — — 1.66% 267.14 12 0.52% 134.58 1.16% 33169.20

A possible explanation for the poor lower bound is as follows. As seen in chapter 4 group-balanced RRTs have a structure requiring bipartite matchings in most periods. Possibly, this structure leads to only slight violations of odd set constraints (6.8) by solutions to the LP relaxation in analogy to the effect observed for HAP-set-based single RRT problems. Then, reduction of solution space by the CG models is small. Table 6.15. Comp. Results for Breaks and Strengths Groups (B&P) |S| = 2 |S| = n2 n l.b. run. t. l.b. run. t. 8 0.00% 496.14 0.48% 889.96 — — — 10 — 12 — — — —

Finally, table 6.15 shows computational results if the minimum number of breaks as well as changing opponents’ strength is required. As seen for changing opponents’ strengths before the lower bound is poor. Hence, run times are higher than those obtained using Cplex. No instances with more than 8 teams can be solved to optimality.

6.7 Summary We propose a B&P approach in order to schedule sports leagues. The approach is quite reasonable since our restricted master problem is a variation of the well known set partitioning problem and our pricing problem is solvable in polynomial time for several cases. Furthermore, our reformulation strengthens the lower bound of the LP relaxation of models given in chapter 3.

6.7 Summary

143

However, results are not encouraging. For almost all instance classes tested we can not compete with run times achieved by employing Cplex. The reason for this might be the rather poor lower bound being more than 2% higher than the one of the LP relaxation only in very exceptional cases. Solely considering instances requiring the minimum number of breaks and having 10 teams we can provide optimal solutions while Cplex fails. However, this might be caused for most parts by the branching scheme developed in section 5.3 and leading to better results if employed in a B&B framework (see section 5.4.2). The B&P approach might perform better for larger problem instances since CG implies the more advantages the more variables are given. On the other hand solving larger instances to optimality seems to be hopeless due to effect of combinatorial explosion observed in the chapter at hand as well as in section 3.3.

7 Conclusions and Outlook

In this work, we consider SLSs with a RRT structure. In chapter 1 we give an overview of related work. Furthermore, we emphasize the meaning of sports leagues and, hence, attractive SLSs. Accordingly, basic optimization and decision problems, respectively, are introduced in chapter 2. Most of these problems are proven to be NP-hard or NP-complete, respectively. Furthermore, we give a strong conjecture concerning the HAP set feasibility problem. Its complexity is open so far and we conjecture it to be solvable in polynomial time. Moreover, IP models representing the problems are introduced. Restricting ourselves to single RRTs we focus on real world requirements in chapter 3. Several common ideas are picked up from literature. Further requirements which have been mostly treated in an abstract way so far are defined in an operational way. First, all requirements are represented by means of IP models. Second, a computational study is carried out using Cplex. Nearly all test runs imply intractability of these problems for relevant numbers of teams. We detect two exceptions, namely HAP-set-based single RRTs and group-balanced RRTs. However, both corresponding optimization problems are proven to be NP-hard in chapters 2 and 4. A promising area of future research might be to vary the IP model formulation for basic problems, first. The lower bound provided by the LP relaxation of model formulations provided in this work is rather weak. Hence, strengthening this bound for basic problems might be a first step towards more encouraging results. Chapter 4 focuses on two concepts introduced as real world requirements: changing and balanced opponents’ strengths. While the basic idea has been stated before the question of operability if a number of teams and a number of strength groups, respectively, are given has not been addressed before (to the best of our knowledge). We answer this

146

7 Conclusions and Outlook

question for almost all cases of n and |S| and give construction schemes for cases with positive answer. Considering future research we propose several directions. First, no construction scheme for group-changing RRTs with n = 6k, k ∈ N+ , and |S| = 3 is known. Second, we suppose the number of teams to be even. Clearly, if n is odd not each team can play in each period. Moreover, if all strength groups have identical sizes n are odd. This case is not considered in the work at |S| as well as |S| hand. Third, RRTs having more than one round are not considered so far. Fourth, in chapter 4 we assume all strength groups to have identical sizes. Generalizing this concept for arbitrary sizes can be considered. In fact, this makes sense in terms of real world tournaments if there are both only a few excellent teams and a few weak teams while the remaining teams can be considered middle level. Fifth, two more concepts concerning strength groups are proposed in chapter 3. Operability for them is open so far. In chapter 5 two branching schemes are developed in order to solve cost minimization problems. They are based on a decomposition scheme where first teams’ venues in each period are fixed and afterwards matches are arranged. Both branching schemes proof to be capable to avoid infeasible subproblems. The branching scheme for the general case is promising but can not compete with Cplex. The one considering RRTs with the minimum number of breaks is clearly superior to Cplex. There are several possible extensions to this concept which might serve as topics for further research. First, we can think of HAP sets having exactly one break per team. These HAP sets are shown to be equivalent to HAP sets having the minimum number of breaks in Miyashiro et al. [64]. Second, we can think of further reducing possible break periods considering stadium availability. Third, the concept of possible break periods seems to be easily adoptable to forbid breaks in certain periods which is a requirement arising in several real world tournaments. In chapter 6 we develop a B&P approach which turned out to be not competitive in comparison to Cplex. We detect two main reasons. First, the lower bound is only slightly better than the one of the LP relaxation to the IP formulations which is rather poor. Second, instances having sizes which would take advantage of the CG approach are beyond tractability. Summarizing, although there has been considerable progress within the last years there still is large potential for further improvements as far as solving RRT optimization problems is concerned. Additionally, many questions from a combinatorial point of view remain to be answered.

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Index

r-RRT (general) 18 r-factorization 58 1-factor 11 1-factorization 2 2-factorization 62

double RRT 2, 12 double RRT (general) 15

attractive matches away-break 84

graph-based models

30

B&P see branch & price beam search 116 binary 1-factorization 60 branch-and-price 101 branching candidate 78 branching scheme 77 branching team 93 breadth first search 83 break 21, 31, 84 break period 85 break sequence 85 break-minimization problem 21 broadcasting station 30 canonical 1-factorization 58 CG see column generation column generation 101 complete graph 2 construction scheme 10 decomposition schemes 18 depth first search 83

economic value 8 edge coloring 2, 9 2

HAP see home-away-pattern home-away-pattern 21 home-away-pattern set 21, 22, 77 home-break 84 infrastructure 29 IP models 2 isomorphic 1-factorizations 11 latin square 2 linear programming relaxation 24 literature 2 match 3 matchday 3 mirrored r-RRT 17 mirrored double RRT 12 multi-graph 62 near-1-factor 58 near-1-factorization 58 opponent schedule 19 ordered 2-factorization 62

156

Index

partial home-away-pattern 79 planar three index assignment problem 5 preference 8, 38 project scheduling problem 2 pseudo-cost 82 PTIAP see planar three index assignment problem round 12 round robin tournament 2 round-based r-RRT 17 round-based double RRT 13 RRT see round robin tournament

security 29 single RRT 2, 6 sports club 1 sports league 3 sports league schedule 1, 3 stadium availability 28 strength group 32 strength group schedule 67 subsequence 92 team 3 venue 3, 77

List of Definitions

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15

PTIAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PTIAP-DEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single RRT Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decision Version of Single RRT Problem . . . . . . . . . . . . . . Mirrored Double RRT Problem . . . . . . . . . . . . . . . . . . . . . . Round-Based Double RRT Problem . . . . . . . . . . . . . . . . . . Double RRT Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirrored rRRT Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . Round-Based r-RRT Problem . . . . . . . . . . . . . . . . . . . . . . . . r-RRT Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Opponent–Schedule Problem . . . . . . . . . . . . . . . . . . . . . . . . . Home-Away-Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Home-Away-Pattern Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HAP set feasibility Problem . . . . . . . . . . . . . . . . . . . . . . . . . HAP–set–based RRT Problem . . . . . . . . . . . . . . . . . . . . . . .

5 6 7 8 12 14 15 17 18 18 19 21 21 22 25

4.1 4.2 4.3 4.4 4.5 4.6

Group-changing single RRT . . . . . . . . . . . . . . . . . . . . . . . . . Group-balanced single RRT . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetric 2-factorization of 2Kk . . . . . . . . . . . . . . . . . . . . . Pairing of Strength Groups . . . . . . . . . . . . . . . . . . . . . . . . . . Group-balanced single RRT problem . . . . . . . . . . . . . . . . . . Group-changing single RRT problem . . . . . . . . . . . . . . . . . .

59 59 64 68 75 76

5.1

Partial Home-Away-Pattern Set . . . . . . . . . . . . . . . . . . . . . . 81

List of Models

2.1 2.2 2.3 2.4 2.5 2.6 2.7

PTIAP-IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SRRTP-IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RBDRRTP-IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DRRTP-IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Opponent-IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HAP-set-feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HAP–set–based SRRTP-IP . . . . . . . . . . . . . . . . . . . . . . . . . .

6 7 14 16 20 23 26

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Forbidden Matches (SRRTP-IP) . . . . . . . . . . . . . . . . . . . . . . Regions (SRRTP-IP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Highly Attended Matches (SRRTP-IP) . . . . . . . . . . . . . . . . Minimum Number of Breaks (SRRTP-IP) . . . . . . . . . . . . . One Break per Team (SRRTP-IP) . . . . . . . . . . . . . . . . . . . . Changing Strength Groups (SRRTP-IP) . . . . . . . . . . . . . . . Balanced Strength Groups (SRRTP-IP) . . . . . . . . . . . . . . . Equally Unchanging Strength Groups (SRRTP-IP) . . . . . Equally Unbalanced Strength Groups (SRRTP-IP) . . . . . Teams’ Preferences (SRRTP-IP) . . . . . . . . . . . . . . . . . . . . . .

31 31 32 33 34 36 37 39 40 41

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

CG CG CG CG CG CG CG CG CG CG

Master – SRRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Master – Minimum Number of Breaks . . . . . . . . . . . . . 107 Master – Changing Strength Groups . . . . . . . . . . . . . . 108 Subproblem – M D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Subproblem – M Dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Subproblem – M D ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Subproblem – M Dpext . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Subproblem – M DpSD . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Subproblem – M DpF F . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Master’s Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

List of Figures

2.1

1-Factorization of K4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 3.2

Run time behavior for single RRT problem . . . . . . . . . . . . 45 Run time behavior for HAP–set–based single RRT problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 4.2 4.3 4.4 4.5

1-Factorization of K4,4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Canonical 1-Factorization of K6 . . . . . . . . . . . . . . . . . . . . . . Binary 1-Factorization of K8 . . . . . . . . . . . . . . . . . . . . . . . . . Binary 1-Factorization of K10 . . . . . . . . . . . . . . . . . . . . . . . . Symmetric 2-factorization of 2K5 . . . . . . . . . . . . . . . . . . . . .

61 61 63 65 66

List of Tables

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Single RRT for n = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirrored double RRT for n = 6 . . . . . . . . . . . . . . . . . . . . . . Round-based double RRT for n = 6 . . . . . . . . . . . . . . . . . . . Double RRT for n = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Opponent Schedule for n = 6 . . . . . . . . . . . . . . . . . . . . . . . . HAP set for n = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infeasible HAP set for n = 14 . . . . . . . . . . . . . . . . . . . . . . . .

6 12 13 15 19 21 24

Opponents’ Strength Groups Not Changing . . . . . . . . . . . . Opponents’ Strength Groups Changing . . . . . . . . . . . . . . . . Opponents’ Strength Groups Not Balanced . . . . . . . . . . . . Opponents’ Strength Groups Balanced . . . . . . . . . . . . . . . . Comp. Results for Single RRT Problem . . . . . . . . . . . . . . . Comp. Results for Forbidden Matches . . . . . . . . . . . . . . . . . Comp. Results for Stadium Availability . . . . . . . . . . . . . . . Comp. Results for HAP–set–based Single RRT Problem . Comp. Results for Highly Attended Matches . . . . . . . . . . . Comp. Results for Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . Comp. Results for Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . Comp. Results for Changing Opponents’ Strength Groups Comp. Results for Balanced Opponents’ Strengths . . . . . . Comp. Results for Equally Unchanging Opponents’ Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Comp. Results for Equally Unbalanced Opponents’ Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Comp. Results for Breaks and Changing Opponents’ Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Comp. Results for Teams’ Preferences . . . . . . . . . . . . . . . . .

35 36 36 38 44 46 47 48 49 50 51 52 53

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

54 54 55 56

164

List of Tables

4.1

Strength group schedule for n = 16, |S| = 4 . . . . . . . . . . . . 69

5.1

Fractional Solution Without Candidate for HAP set branching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Example for 3 HAPs with too few matches . . . . . . . . . . . . 89 5.3 Example for ordered HAP set and equivalent representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 T ′ before (left) and after (right) circulating . . . . . . . . . . . . 93 5.5 Equivalent representation of T ′ (left) and T ′′ (right) . . . . 93 5.6 T ′′ before (left) and after (right) recirculating . . . . . . . . . . 93 5.7 Infeasible subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.8 Comp. Results for Branching on General HAP Sets . . . . . 99 5.9 Comp. Results for Branching Strategies . . . . . . . . . . . . . . . 100 5.10 Comp. Results for Minimum Number of Breaks . . . . . . . . 101 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15

Dual Variables of CG Master . . . . . . . . . . . . . . . . . . . . . . . . 108 Fractional Solution Without Candidate Same/Differ . . . . 115 Fractional Solution Without Candidate Fix/Forbid . . . . . 116 Lower bounds for CG master problem according to reduced cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Reduction of run times per node by lower bounds . . . . . . 133 Key figures for upper bound heuristic . . . . . . . . . . . . . . . . . 136 Comp. Results for Single RRT problem (B&P) . . . . . . . . . 138 Comp. Results for Forbidden Matches (B&P) . . . . . . . . . . 138 Comp. Results for Stadium Availability (B&P) . . . . . . . . . 139 Comp. Results for HAP–set–based Single RRT Problem (B&P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Comp. Results for Highly Attended Matches (B&P) . . . . 140 Comp. Results for Regions (B&P) . . . . . . . . . . . . . . . . . . . . 141 Comp. Results for Breaks (B&P) . . . . . . . . . . . . . . . . . . . . . 141 Comp. Results for Changing Opponents’ Strengths (B&P)142 Comp. Results for Breaks and Strengths Groups (B&P) . 142

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Vol. 569: L. Neubecker, Strategic Competition in Oligopolies with Fluctuating Demand. IX, 233 pages. 2006. Vol. 570: J. Woo, The Political Economy of Fiscal Policy. X, 169 pages. 2006. Vol. 571: T. Herwig, Market-Conform Valuation of Options. VIII, 104 pages. 2006. Vol. 572: M. F. Jäkel, Pensionomics. XII, 316 pages. 2006 Vol. 573: J. Emami Namini, International Trade and Multinational Activity. X, 159 pages, 2006. Vol. 574: R. Kleber, Dynamic Inventory Management in Reverse Logistics. XII, 181 pages, 2006. Vol. 575: R. Hellermann, Capacity Options for Revenue Management. XV, 199 pages, 2006. Vol. 576: J. Zajac, Economics Dynamics, Information and Equilibnum. X, 284 pages, 2006. Vol. 577: K. Rudolph, Bargaining Power Effects in Financial Contracting. XVIII, 330 pages, 2006. Vol. 578: J. Kühn, Optimal Risk-Return Trade-Offs of Commercial Banks. IX, 149 pages, 2006. Vol. 579: D. Sondermann, Introduction to Stochastic Calculus for Finance. X, 136 pages, 2006. Vol. 580: S. Seifert, Posted Price Offers in Internet Auction Markets. IX, 186 pages, 2006. Vol. 581: K. Marti; Y. Ermoliev; M. Makowsk; G. Pflug (Eds.), Coping with Uncertainty. XIII, 330 pages, 2006. Vol. 582: J. Andritzky, Sovereign Default Risks Valuation: Implications of Debt Crises and Bond Restructurings. VIII, 251 pages, 2006. Vol. 583: I.V. Konnov, D.T. Luc, A.M. Rubinov† (Eds.), Generalized Convexity and Related Topics. IX, 469 pages, 2006. Vol. 584: C. Bruun, Adances in Artificial Economics: The Economy as a Complex Dynamic System. XVI, 296 pages, 2006. Vol. 585: R. Pope, J. Leitner, U. Leopold-Wildburger, The Knowledge Ahead Approach to Risk. XVI, 218 pages, 2007 (planned). Vol. 586: B.Lebreton, Strategic Closed-Loop Supply Chain Management. X, 150 pages, 2007 (planned). Vol. 587: P. N. Baecker, Real Options and Intellectual Property: Capital Budgeting Under Imperfect Patent Protection. X, 276 pages , 2007.

Vol. 592: A. C.-L. Chian, Complex Systems Approach to Economic Dynamics. X, 95 pages, 2007. Vol. 593: J. Rubart, The Employment Effects of Technological Change: Heterogenous Labor, Wage Inequality and Unemployment. XII, 209 pages, 2007 Vol. 594: R. Hübner, Strategic Supply Chain Management in Process Industries: An Application to Specialty Chemicals Production Network Design. XII, 243 pages, 2007 Vol. 595: H. Gimpel, Preferences in Negotiations: The Attachment Effect. XIV, 268 pages, 2007 Vol. 596: M. Müller-Bungart, Revenue Management with Flexible Products: Models and Methods for the Broadcasting Industry. XXI, 297 pages, 2007 Vol. 597: C. Barz, Risk-Averse Capacity Control in Revenue Management. XIV, 163 pages, 2007 Vol. 598: A. Ule, Partner Choice and Cooperation in Networks: Theory and Experimental Evidence. Approx. 200 pages, 2007 Vol. 599: A. Consiglio, Artificial Markets Modeling: Methods and Applications. XV, 277 pages, 2007 Vol. 600: M. Hickman, P. Mirchandani, S. Voss (Eds.): Computer-Aided Scheduling of Public Transport. Approx. 424 pages, 2007 Vol. 601: D. Radulescu: CGE Models and Capital Income Tax Reforms: The Case of a Dual Income Tax for Germany. XVI, 168 pages, 2007 Vol. 602: N. Ehrentreich: Agent-Based Modeling: The Santa Fe Institute Artificial Stock Market Model Revisited. XVI, 225 pages, 2007 Vol. 603: D. Briskorn, Sports Leagues Scheduling: Models, Combinatorial Properties, and Optimization Algorithms. XII, 164 pages, 2008 Vol. 604: D. Brown, F. Kubler, Computational Aspects of General Equilibrium Theory: Refutable Theories of Value. XII, 202 pages, 2008 Vol. 605: M. Puhle, Bond Portfolio Optimization. XIV, 137 pages, 2008 Vol. 606: S. von Widekind, Evolution of Non-Expected Utility Preferences. X, 130 pages, 2008 Vol. 607: M. Bouziane, Pricing Interest Rate Derivatives: A Fourier-Transform Based Approach. XII, 191 pages, 2008

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Generalized Convexity and Optimization: Theory and Applications (Lecture Notes in Economics and Mathematical Systems)

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Conjugate Duality in Convex Optimization (Lecture Notes in Economics and Mathematical Systems)

Artificial Economics: The Generative Method in Economics (Lecture Notes in Economics and Mathematical Systems)

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