Vertex Cover Resists SDPs Tightened by Local Hypermetric Inequalities

We consider the standard semidefinite programming (SDP) relaxation for the vertex cover problem to which all hypermetric inequalities supported on at most k vertices are added and show that the integrality gap for such SDPs remains 2− o(1) even for k = O( √ logn/ log log n). This extends results by Kleinberg-Goemans, Charikar and Hatami et al. who considered vertex cover SDPs tightened using the triangle and pentagonal inequalities, respectively. Our result is complementary to a recent result by Georgiou et al. proving integrality gaps for vertex cover SDPs in the Lovász-Schrijver hierarchy. However, the SDPs we consider are incomparable to the SDPs analyzed by Georgiou et al. In particular we show that vertex cover SDPs in the Lovász-Schrijver hierarchy fail to satisfy any hypermetric constraints supported on an independent set of the input graph. This constrasts with the LP Lovász-Schrijver hierarchy where all local LP constraints are derived. ∗Funded in part by NSERC