lec16 - CS151 Complexity Theory Lecture 16 May 19, 2011 New...

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CS151 Complexity Theory Lecture 16 May 19, 2011
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May 19, 2011 2 New topic(s) Optimization problems, Approximation Algorithms, and Probabilistically Checkable Proofs
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May 19, 2011 3 Approximation Algorithms “gap-producing” reduction from NP - complete problem L 1 to L 2 no yes L 1 L (min. problem) f opt k rk
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May 19, 2011 4 Gap producing reductions r-gap-producing reduction : f computable in poly time – x L 1 opt(f(x)) k – x L 1 opt(f(x)) > rk for max. problems use “ k ” and “ < k/r Note: target problem is not a language promise problem (yes no not all strings) “promise”: instances always from (yes no)
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May 19, 2011 5 MAX-k-SAT Missing link: first gap-producing reduction history’s guide it should have something to do with SAT Definition: MAX-k-SAT with gap ε instance: k-CNF φ YES: some assignment satisfies all clauses NO: no assignment satisfies more than (1 – ε ) fraction of clauses
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May 19, 2011 6 Proof systems viewpoint MAX-k-SAT with gap ε NP -hard for any language L NP proof system of form: – given x, compute reduction to MAX-k-SAT: ϕ x – expected proof is satisfying assignment for ϕ x verifier picks random clause (“local test”) and checks that it is satisfied by the assignment x L Pr[verifier accepts] = 1 x L Pr[verifier accepts] ≤ (1 – ε) can repeat O(1/ε) times for error < ½
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May 19, 2011 7 Proof systems viewpoint can think of reduction showing k-SAT NP-hard as designing a proof system for NP in which: verifier only performs local tests can think of reduction showing MAX-k-SAT with gap ε NP-hard as designing a proof system for NP in which: verifier only performs local tests invalidity of proof* evident all over: “holographic proof” and an ε fraction of tests notice such invalidity
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May 19, 2011 8 PCP Probabilistically Checkable Proof (PCP) permits novel way of verifying proof: pick random local test query proof in specified k locations accept iff passes test fancy name for a NP-hardness reduction
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May 19, 2011 9 PCP PCP[r(n),q(n)] : set of languages L with p.p.t. verifier V that has (r, q)-restricted access to a string “proof” V tosses O(r(n)) coins V accesses proof in O(q(n)) locations ( completeness ) x L 5 proof such that Pr[V(x, proof) accepts] = 1 ( soundness ) x L 2200 proof* Pr[V(x, proof*) accepts] ½
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May 19, 2011 10 PCP Two observations: PCP[1, poly n] = NP proof? PCP[log n, 1] NP proof? The PCP Theorem (AS, ALMSS): PCP[log n, 1] = NP .
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May 19, 2011 11 PCP Corollary : MAX-k-SAT is NP -hard to approximate to within some constant ε . using PCP[log n, 1] protocol for, say, VC
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lec16 - CS151 Complexity Theory Lecture 16 May 19, 2011 New...

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