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# lec6 - Clique CLIQUE = cfw(G k | G is a graph with a clique...

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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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