lecture23 - CSci 5403 COMPLEXITY THEORY LECTURE XXIII: PCPs...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XXIII: PCPs AND HARDNESS OF APPROXIMATION Definition. A gap problem is a promise problem ( Π Y , Π N ) derived from optimization problem (R,val): Π Y = { x : OPT(x) c }, Π N = {x : OPT(x) < c/ ρ }. Claim. If there is a reduction from 3SAT to ( Π Y , Π N ) Then there is no c/ ρ approximation algorithm for (R,val) unless P=NP. Definition. MAX-INDSET = (R,val) where R = {((V,E), S) : S ˧ V and ˲ u,v ˥ S, (u,v) E } val(S) = |S|. Theorem. ˲ c > 0 : unless P=NP there is no c-approximation scheme for MAX-INDSET. Lemma. There exists a reduction ƒ from 3CNFs to graphs such that OPT INDSET (ƒ( ϕ )) = OPT MAX3SAT ( ϕ ). Sketch. Consider ϕ =(x 1 ˮ x 2 ) ˭ ( ͞ x 1 ˮ ͞ x 3 ) 01 11 10 01 00 10 x satisfying k clauses ind-set S of size k Put an edge between two nodes iff they have inconsistent assignments. Lemma. ˲ ρ > 0: There is a (c, ρ ) (c 2 , ρ 2 ) gap preserving reduction from INDSET to INDSET. (This is called an amplifier) Proof. For graph G = (V,E), let G 2 be the graph (V × V,E’) where E’ is: { ʪ (u ,u ),(v ,v ) ʫ | (u =v (u ,v ) ˥ E) ˮ (u ,v ) ˥ E}). Then max(G) = k ˱ max(G ² ) = k ² . Proof. (of Theorem) Sufficient composition of amplifier to gap-preserving MAX3SAT reduction.
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2 Definition. Oracle PPTM V is an (r,q)-verifier for language L if 1. ˲ O,x. V O (x) uses at most r(|x|) random bits and makes at most q(|x|) oracle queries. 2. x ˥ L ˰ ˳ π ˥ {0,1}*: Pr r [V π (x;r)=1] = 1. (Completeness) 3. x L ˰ ˲ π ˥ {0,1}*, Pr r [V π (x;r)=1] ½ (Soundness) Definition. L ˥ PCP[r(n),q(n)] iff there is a (r,q) verifier for L. Theorem.
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lecture23 - CSci 5403 COMPLEXITY THEORY LECTURE XXIII: PCPs...

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