lec6 - CS151 Complexity Theory Lecture 6 Clique CLIQUE =...

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CS151 Complexity Theory Lecture 6 April 14, 2011

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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

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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)

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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
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

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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)

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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 ( 29 ÷ ÷ 1/ 4 1/ 4 n 1/ 4 n 4 2 1/ n n 2 n n
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:

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lec6 - CS151 Complexity Theory Lecture 6 Clique CLIQUE =...

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