Extending SDP integrality gaps to Sherali-Adams with applications to Quadratic Programming and MaxCutGain

Abstract

We show how under certain conditions one can extend constructions of integrality gaps for semidefinite relaxations into ones that hold for stronger systems: those SDP to which the so-called k-level constraints of the Sherali-Adams hierarchy are added. The value of k above depends on properties of the problem. We present two applications, to the Quadratic Programming problem and to the MaxCutGain problem. Our technique is inspired by a paper of Raghavendra and Steurer [Raghavendra and Steurer, FOCS 09] and our result gives a doubly exponential improvement for Quadratic Programming on another result by the same authors [Raghavendra and Steurer, FOCS 09]. They provide tight integrality-gap for the system above which is valid up to k = (log log n) Ω(1) whereas we give such a gap for up to k = n Ω(1). © 2010 Springer-Verlag.