Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-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 technicptes 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. Lovasz 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 Cur and Sparsest Cur mentioned above. We rule out polynomial-time 2 - Omega(lfloor) approximations for Vertex Cover using LS₊. In particular, we prove an integrality gap of 2 - o(lfloor)for Vertex Cover SDPs obtained by tightening the standard LP relaxation with Omega(radiclog n/ log log n) rounds of LS₊. While tight integrality gaps were known for Vertex Cover in the weaker LS system [23 ], previous results did not rule out a2 - Omega(1) approximation after even two rounds of LS₊.