# NPC - N PC M or e NP-Compl ete Pr obl ems Complexity...

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Complexity ©D.Moshko 1 NPC More NP-Complete Problems

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Complexity ©D.Moshko 2 Introduction Objectives : To introduce more NP-Complete problems. Overview : 3SAT CLIQUE & INDEPENDENT-SET SUBSET-SUM
Complexity ©D.Moshko 3 Method How to show a problem is in NPC ? First show it’s in NP Then show it is NP-hard by reducing some NP-Hard problem to it.

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Complexity ©D.Moshko 4 New Base Problems The only NP-Complete problem we currently know of is SAT . Unfortunately, it’s not very comfortable to work with. Thus we’ll start by introducing several useful variants of . We’ll use them as our base problems.
Complexity ©D.Moshko 5 3SAT Instance : a 3CNF formula Problem : To decide if the formula is satisfiable. (x y ∨¬ z) (x y (x x x) ( ¬ x x x) A satifiable 3CNF formula An unsatifiable 3CNF formula

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Complexity ©D.Moshko 6 3SAT is NP-Complete 3SAT is a special case of SAT , and is therefore clearly in NP . In order to show it’s also NP-Complete , we’ll alter the proof of SAT ’s NP-Completeness , so it produces 3CNF formulas. Why would that be enough? SIP 259-260
Complexity ©D.Moshko 7 Revisiting SAT’s NP- Completeness Proof = ) x x ( x φ t j, i, s j, i, C t s s j, i, C s n j i, 1 cell k # , n 1, _ n _ 2, n w 1,3, q 1,2, # 1,1, start k k 1 0 x x ... x x x x φ = - + accept k q j, i, n j i, 1 x φ = ( 29 k 1 6 1 6 move i 1,j ,a i 1,j 1,a 1 i,j n legal a ,. ..,a x x φ - + + = Given a TM and an input we’ve produced a conjunction of:

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Complexity ©D.Moshko 8 Transforming the Formula into a CNF Formula All the sub-formulas, but φ move , form a CNF formula.
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NPC - N PC M or e NP-Compl ete Pr obl ems Complexity...

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