Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász–Schrijver Hierarchy

Abstract

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. Lovasz 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 $2-\Omega(1)$ approximations for Vertex Cover using $LS_+$. In particular, for every $\epsilon>0$ we prove an integrality gap of $2-\epsilon$ for Vertex Cover SDPs obtained by tightening the standard LP relaxation with $\Omega(\sqrt{\log n/\log\log n})$ 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 $2-\Omega(1)$ approximation after even two rounds of $LS_+$.