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 max-cut, 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. We survey results that built upon this work.

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.

Lovász and Schrijver [13] defined three progressively stronger procedures LS 0 , LS and LS + , for systematically tightening linear relaxations over many rounds. All three procedures yield the integral hull after at most n rounds. On the other hand, constant rounds of LS + can derive the relaxations behind many famous approximation algorithms such as those for MAX-CUT, SPARSEST-CUT. So proving round lower bounds for these procedures on specific problems may give evidence about inapproximability. We prove new round lower bounds for VERTEX COVER in the LS hierarchy. Arora et al. [3] showed that the integrality gap for VERTEX COVER relaxations remains 2−o(1) even after Ω(log n) rounds LS. However, their method can only prove round lower bounds as large as the girth of the input graph, which is O(log n) for interesting graphs. We break through this "girth barrier" and show that the integrality gap for VERTEX COVER remains 1.5 − even after Ω(log 2 n) rounds of LS. In contrast, the best PCPbased results only rule out 1.36-approximations. Moreover, we conjecture that the new technique we introduce to prove our lower bound, the "fence" method, may lead to linear or nearly linear LS round lower bounds for VERTEX COVER.

are generalized and combined with an argument for diagonalizing over machines taking n bits of advice on inputs of length n to obtain the first nontrivial time-space lower bounds for SAT on nonuniform machines.In particular, we show that for any a <`2 and any e > 0, SAT cannot be computed by a random access deterministic Turing machine using n a time, ) is properly contained in NTIME(n r ).