lec6 - Clique CLIQUE = cfw_ (G, k) | G is a graph with a...

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1 CS151 Complexity Theory Lecture 6 April 14, 2011 April 14, 2011 2 Clique CLIQUE = { (G, k) | G is a graph with a clique of size ≥ k } (clique = set of vertices every pair of which are connected by an edge) • CLIQUE is NP -complete. April 14, 2011 3 Circuit lower bounds • We think that NP requires exponential-size circuits. • Where should we look for a problem to attempt to prove this? • Intuition: “hardest problems” – i.e., NP - complete problems April 14, 2011 4 Circuit lower bounds • Formally: – if any problem in NP requires super- polynomial size circuits – then every NP -complete problem requires super-polynomial size circuits – Proof idea : poly-time reductions can be performed by poly-size circuits using a variant of CVAL construction April 14, 2011 5 Monotone problems • Definition: monotone language = language L {0,1} * such that x L implies x’ L for all x ¹ x’. – flipping a bit of the input from 0 to 1 can only change the output from “no” to “yes” (or not at all) April 14, 2011 6 Monotone problems • some NP -complete languages are monotone – e.g. CLIQUE (given as adjacency matrix): – others: HAMILTON CYCLE, SET COVER – but not SAT, KNAPSACK
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2 April 14, 2011 7 Monotone circuits A restricted class of circuits: • Definition: monotone circuit = circuit whose gates are ANDs ( ), ORs ( ), but no NOTs • can compute exactly the monotone fns. – monotone functions closed under AND, OR April 14, 2011 8 Monotone circuits • A question: Do all poly-time computable monotone functions have poly-size monotone circuits? – recall: true in non-monotone case April 14, 2011 9 Monotone circuits A monotone circuit for CLIQUE n,k • Input: graph G = (V,E) as adj. matrix, |V|=n – variable x i,j for each possible edge (i,j) • ISCLIQUE(S) = monotone circuit that = 1 iff S V is a clique: i,j S x i,j • CLIQUE n,k computed by monotone circuit: S V, |S| = k ISCLIQUE(S) April 14, 2011 10 Monotone circuits • Size of this monotone circuit for CLIQUE n,k : • when k = n 1/4 , size is approximately: n k k 2 1/4 n n 4 2 1/ n n 2 n n April 14, 2011 11 Monotone circuits • Theorem (Razborov 85): monotone circuits for CLIQUE n,k with k = n 1/4 must have size at least 2 Ω(n 1/8 ) . • Proof: – rest of lecture April 14, 2011 12 Proof idea • “method of approximation” • suppose C is a monotone circuit for CLIQUE n,k • build another monotone circuit CC that “approximates” C gate-by-gate
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3 April 14, 2011 13
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lec6 - Clique CLIQUE = cfw_ (G, k) | G is a graph with a...

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