MIT6_045JS11_lec15

# MIT6_045JS11_lec15 - 6.045 Automata Computability and...

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6.045: Automata, Computability, and Complexity (GITCS) Class 15 Nancy Lynch

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Today: More Complexity Theory • Polynomial-time reducibility, NP-completeness, and the Satisfiability (SAT) problem • Topics: – Introduction (Review and preview) – Polynomial-time reducibility, p – Clique p VertexCover and vice versa – NP-completeness – SAT is NP-complete • Reading: – Sipser Sections 7.4-7.5 •N e x t : – Sipser Sections 7.4-7.5
Introduction

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Introduction •P = { L | there is some polynomial-time deterministic Turing machine that decides L } •N P = { L | there is some polynomial-time nondeterministic Turing machine that decides L } • Alternatively, L NP if and only if ( V, a polynomial-time verifier ) ( p, a polynomial ) such that: x L iff ( c, |c| p(|x|) ) [ V( x, c ) accepts ] • To show that L NP, we need only exhibit a suitable verifier V and show that it works (which requires saying what the certificates are). •P NP, but it’s not known whether P = NP. certificate
Introduction •P = { L | poly-time deterministic TM that decides L } •N P = { L | poly-time nondeterministic TM that decides L } •L NP if and only if ( V, poly-time verifier ) ( p, poly) x L iff ( c, |c| p(|x|) ) [ V( x, c ) accepts ] • Some languages are in NP, but are not known to be in P (and are not known to not be in P ): – SAT = { < φ > | φ is a satisfiable Boolean formula } –3COLOR = { < G > | G is an (undirected) graph whose vertices can be colored with 3 colors with no 2 adjacent vertices colored the same } –CL IQUE = { < G, k > | G is a graph with a k-clique } – VERTEX-COVER = { < G, k > | G is a graph having a

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CLIQUE • CLIQUE = { < G, k > | G is a graph with a k-clique } • k-clique: k vertices with edges between all pairs in the clique. • In NP, not known to be in P, not known to not be in P. • 3-cliques: { b, c, d }, { c, d, f } • Cliques are easy to verify, but may be hard to find. a b c d e f
CLIQUE • CLIQUE = { < G, k > | G is a graph with a k-clique } • Input to the VC problem: < G, 3 > • Certificate, to show that < G, 3 > CLIQUE, is { b, c, d } (or { c, d, f }). • Polynomial-time verifier can check that { b, c, d } is a 3-clique. a b c d e f

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= { < G, k > | G is a graph with a vertex cover of size k } • Vertex cover of G = (V, E): A subset C of V such that, for every edge (u,v) in E, either u C or v C. – A set of vertices that “covers” all the edges. • In NP, not known to be in P, not known to not be in P. • 3-vc: { a, b, d } • Vertex covers are easy to verify, may be hard to find. a
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• Spring '11
• Prof.ScottAaronson
• Computational complexity theory, Clique, NP-complete, Boolean satisfiability problem, polynomial-time reducibility

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MIT6_045JS11_lec15 - 6.045 Automata Computability and...

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