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l13-handout - Outline NP-completeness Lecture 13 Proving...

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Lecture 13: Proving problems are NP-Complete David Dill Department of Computer Science 1 Outline NP-completeness SAT CSAT 3SAT 2 NP-complete problems This is about decision problems (problems with yes/no answers). Equivalently, solving the membership problem x L . The class P – problems that are solvable in polynomial time on an DTM. (P-time, polynomial time). The class NP – problems that are solvable in polynomial time on an NTM. Obviously P NP . No one knows whether NP P (the famous P = NP problem). NP-complete problems are the “hardest” problems in NP. If there are any problems in NP - P , the NP-complete problems are all there. Every NP-complete problem can be translated in deterministic polynomial time to every other NP-complete problem. So, if there is a P-time to one NP-complete problem, there is a P-time solution to every P-time problem. 3 NP-hardness by reduction Typical method: Reduce a known NP-hard problem P 1 to the new problem P 2 . A reduction is a polynomial-time translation of the problem, call it r . If w is an instance of problem P 1 , then r ( w ) is an instance of problem P 2 . r must have two properties: it preserves the answer. So the answer to w is “yes” iff the answer to r ( w ) is “yes.” r ( w ) can be computed in time polynomial in | w | . 4
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Basic Proof Strategy NP-completeness is a good news/bad news situation.
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