{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec16

# lec16 - CS151 Complexity Theory Lecture 16 New topic(s...

This preview shows pages 1–12. Sign up to view the full content.

CS151 Complexity Theory Lecture 16 May 19, 2011

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
May 19, 2011 2 New topic(s) Optimization problems, Approximation Algorithms, and Probabilistically Checkable Proofs
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 38

lec16 - CS151 Complexity Theory Lecture 16 New topic(s...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online