We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [23] 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 Ω(n) rounds of the Lasserre hierarchy. We show that for the general MAX k-CSP problem, this ratio can be as large as 2 k /2k − when the alphabet is binary and q k /q(q − 1)k − when the alphabet size a prime q, even after Ω(n) rounds of the Lasserre hierarchy. 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 Independent Set and Chromatic Number, we show integrality gaps of n/2
Via reductions from this result, Tulsiani showed gap instances for Max k-CSP (for Ω(n) rounds), and instances with n 1−o(1) gap for ≈ 2 √ log n rounds for the Independent Set and Chromatic Numbers REF .
CSP gaps and reductions in the lasserre hierarchy
STOC '09
Mathematics Computer Science