Integrality gaps of 2 − o(1) for Vertex Cover SDPs in the Lovász-Schrijver hierarchy

Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for MAX CUT and SPARSEST CUT use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is VERTEX COVER. PCP-based techniques of Dinur and Safra [7] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. Furthermore, there is a widespread belief that SDP techniques are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [3], our aim is to show that a large family of LP and SDP based algorithms fail to produce an approximation for VERTEX COVER better than 2. Lovász and Schrijver [21] introduced the systems LS and LS+ for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, LS+ captures the celebrated SDP-based algorithms for MAX CUT and SPARSEST CUT mentioned above. We rule out polynomial-time 2 − Ω(1) approximations for VERTEX COVER using LS+. In particular, we prove an integrality gap of 2−o(1) for VERTEX COVER SDPs obtained by tightening the standard LP relaxation with Ω( √ logn/ log logn) rounds of LS+. While tight integrality gaps were known for VERTEX COVER in the weaker LS system [23], previous results did not rule out Funded in part by NSERC a 2−Ω(1) approximation after even two rounds of LS+.