This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
EF, and therefore Ax accepts no inputs, which implies that DecideX(x) will find no integer i such that TJ (Ax) is satisfiable. This completes the proof that X
4, let 11, 12, .. , lk be the literals such that ' 1l + 12 + +k . Let u l, u 2 , . . Uk-3 be new Boolean variables. Take = (11 +12+U 1 )
If the expression ' consists of several clauses, treat each of them independently (using different new variables for each clause) and form the conjunction of all the expressions in 3-CNF thus obtained. Example 10.3.12.
(Continuation of Example 10.3.11)
' = (p +q +r +s)(r-is)(f +s +x +v +w ) we obtain = (p
+q+ul)(uIX+r +s)(r +s)(p +s +u2 )(u 2 +x +U3)(u 3 +V +W )
Introduction to Complexity
Problem 10.3.23. Prove that , is satisfiable if and only if ' the construction of Example 10.3.11.
is satisfiable in 0
Problem 10.3.24. Prove that SAT-CNF < P SAT-3-CNF still holds even if in the definition of SAT-3-CNF we insist that each clause should contain exactly three literals. 0 Example 10.3.13. 3-COL is NP-complete. Let G be an undirected graph and k an integer constant. The problem k-COL consists of determining whether G can be coloured with k colours (see Problem 10.3.6). It is easy to see that 3-COLeNP. To show that 3-COL is NP-complete, we shall prove this time that SAT-3-CNF