Approximating CSPs Using LP Relaxation

This paper studies how well the standard LP relaxation ap-proximates a k-ary constraint satisfaction problem (CSP) on label set[L]. We show that, assuming the Unique Games Conjecture, it achievesan approximation within O(k · logL) of the optimal approximation fac-tor. In particular we prove the following hardness result: let I be a k-aryCSP on label set [L] with constraints from a constraint class C, such thatit is a (c, s)-integrality gap for the standard LP relaxation. Then, givenan instance H with constraints from C, it is NP-hard to decide whether, opt(H) ≥ Ω(ck3 logL), or opt(H) ≤ 4 · s, assuming the Unique Games Conjecture. We also show the existence of anefficient LP rounding algorithm Round such that a lower bound for it canbe translated into a similar (but weaker) hardness result. In particular,if there is an instance from a permutation invariant constraint class Cwhich is a (c, s)-rounding gap for Round, then given an instance H withconstraints from C, it is NP-hard to decide whether, opt(H) ≥ Ω(ck3 logL), or opt(H) ≤ O((logL))· s, assuming the Unique Games Conjecture.