Vertex Cover Resists SDPs Tightened by Local Hypermetric Inequalities

Abstract

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( log n/ 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.

Local versus global properties of metric spaces

“…It turns out that the better (as a function of k) such metric classes approximate the original 1 cone, the faster should the related lift-and-project procedure converge to the optimum. We address the interested reader to [3,16].…”

“…Hence, we may replace distortion with the (supremum) stretch incurred by d. It will be convenient to bring the discussion back to the realm of discrete metric spaces. Instead of proving (16) for R n , we shall prove it for Z n . Clearly, this is a fully equivalent statement (by scaling and taking limits).…”

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

“…Moreover, the vectors satisfy the extended triangle inequalities (3) employed by Karakostas's SDP. Interestingly, the asymptotic bound for the parameter k in [11] is the same as the number of LS + rounds for which we prove our lower bound in the current paper. This hints at a deeper relationship between the families of SDPs considered in [11] and the current paper.…”

“…Partial progress along this line is made in [11] where it is shown that the construction from the current paper (modulo an affine transformation) satisfies VERTEX COVER SDPs tightened by local hypermetric inequalities (hypermetric inequalities are a canonical subfamily of inequalities satisfied by all 1 metrics and include triangle, pentagonal and indeed all (2k + 1)-gonal inequalities). More precisely, the SDP solution analyzed in [11] arises by taking the Cholesky decomposition of the first-round protection matrix from the current paper and then applying the affine transformation z i = 2v 0 − v i (this simply maps {0, 1} integral solutions to {1, −1}).…”

“…More precisely, the SDP solution analyzed in [11] arises by taking the Cholesky decomposition of the first-round protection matrix from the current paper and then applying the affine transformation z i = 2v 0 − v i (this simply maps {0, 1} integral solutions to {1, −1}). It is shown in [11] that the 2 2 metric induced by these vectors satisfies all k-gonal inequalities for k = Ω( log n/ log log n) (and actually satisfies all hypermetric inequalities on k points). Moreover, the vectors satisfy the extended triangle inequalities (3) employed by Karakostas's SDP.…”