Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy

Abstract

Lovász and Schrijver described a generic method of tightening the LP and SDP relaxation for any 0-1 optimization problem. These tightened relaxations were the basis of several celebrated approximation algorithms (such as for maxcut, max-3sat, and sparsest cut).We prove strong inapproximability results in this model for well-known problems such as max-3sat, hypergraph vertex cover and set cover. We show that the relaxations produced by as many as Ω(n) rounds of the LS+ procedure do not allow nontrivial approximation, thus ruling out the possibility that the LS+ approach gives even slightly subexponential approximation algorithms for these problems.We also point out why our results are somewhat incomparable to known inapproximability results proved using PCPs, and formalize several interesting open questions.

Goldreich's PRG: Evidence for Near-Optimal Polynomial Stretch

“…In order to prove security against SDP hierarchies, we will need H − S to have high expansion for sets S such that |S| ≤ r for some r. This is not true in general, but [26], [23], [25] give an algorithm for finding a superset S of S such thatS is not too much bigger than S and H −S has high expansion:…”

Linear Level Lasserre Lower Bounds for Certain k-CSPs

“…Buresh-Oppenheim, Galesy, Hoory, Magen and Pitassi [BOGH + 03], and Alekhnovich, Arora, Tourlakis [AAT05] prove Ω(n) LS+ round lower bounds for proving the unsatisfiability of random instances of 3-SAT (and, in general, k-SAT with k ≥ 3) and Ω(n) 2 round lower bounds for achieving approximation factors better than 7/8 − ε for Max 3-SAT, better than (1 − ε) ln n for Set Cover, and better than k − 1 − ε for HypergraphVertexCover in k-uniform hypergraphs. They leave open the question of proving LS+ round lower bounds for approximating the Vertex Cover problem.…”

“…A more complete comparison can be found in [Lau03]. While there have been a growing number of integrality gap lower bounds for the LS [ABL02,ABLT06,Tou06,STT07b], the LS+[BOGH + 03, AAT05,STT07a,GMPT06], and the SA [dlVKM07,CMM07] hierarchies, similar bounds for the Lasserre hierarchy have remained elusive.…”

Untitled

“…Now we prove that the incidence graph of a hypergraph sampled from H k (m, n, n 0 , Γ) is locally sparse with high probability. The proof of the following lemma follows closely proofs of analogous statements in [1,6]. Proof.…”

“…In particular, they proved that when the condition in their characterization is satisfied, there exists a (1 − o(1), ρ( f ) + o(1))-integrality gap for O(log log n) levels of Sherali-Adams hierarchy for predicates f . Here, we show that using Theorem 1.1, their result can be simplified and strengthened 1 to O (log n/ log log n) levels.…”