CSP gaps and reductions in the lasserre hierarchy

We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX k-XOR, the ratio of the SDP optimum to the integer optimum may be as large as 2 even after (n) rounds of the Lasserre hierarchy. We show that for the general MAX k-CSP problem over binary domain, the ratio of SDP optimum to the value achieved by the optimal assignment, can be as large as 2k /2k - even after (n) rounds of the Lasserre hierarchy. For alphabet size q which is a prime, we give a lower bound of qk /q(q - 1)k - for (n) rounds. The method of proof also gives optimal integrality gaps for a predicate chosen at random. We also explore how to translate gaps for CSP into integrality gaps for other problems using reductions, and establish SDP gaps for Maximum Independent Set, Approximate Graph Coloring, Chromatic Number and Minimum Vertex Cover. For Indep...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Madhur Tulsiani
• Keywords: Algorithms, Integrality Gaps, Optimal Integrality Gaps, SDP Gaps, STOC 2009