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. Probabilistically checkable proof (PCP)-based techniques of Dinur and Safra [Ann. of Math. (2), 162 (2005), pp. 439–486] 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. There is a widespread belief that semidefinite programming (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. [Theory Comput., 2 (2006), pp. 19–51], our aim is to show that a large family of linear programming (LP)and SDP-based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] 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 SDP-based 2 −Ω(1) approximations for Vertex Cover using LS+. In particular, for every > 0 we prove an integrality gap of 2− 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 [G. Schoenebeck, L. Trevisan, and M. Tulsiani, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2007, pp. 302–310], previous results did not rule out a 2−Ω(1) approximation after even two rounds of LS+.