Approximating CSPs Using LP Relaxation

Abstract

This paper studies how well the standard LP relaxation approximates a $k$-ary constraint satisfaction problem (CSP) on label set $[L]$. We show that, assuming the Unique Games Conjecture, it achieves an approximation within $O(k^3\cdot \log L)$ of the optimal approximation factor. In particular we prove the following hardness result: let $\mathcal {I}$ be a $k$-ary CSP on label set $[L]$ with constraints from a constraint class $\mathcal {C}$, such that it is a $(c,s)$-integrality gap for the standard LP relaxation. Then, given an instance $\mathcal {H}$ with constraints from $\mathcal {C}$, it is NP-hard to decide whether,

$$\mathsf{opt} (\mathcal {H}) \ge \varOmega \left( \frac{c}{k^3\log L}\right) ,\ \ \text { or }\ \ \mathsf{opt} (\mathcal {H}) \le 4\cdot s,$$

assuming the Unique Games Conjecture. We also show the existence of an efficient LP rounding algorithm $\mathsf{Round}$ such that given an instance $\mathcal {H}$ from a permutation invariant constraint class $\mathcal {C}$ which is a $(c,s)$-rounding gap for $\mathsf{Round}$, it is NP-hard to decide whether,

$$\mathsf{opt} (\mathcal {H}) \ge \varOmega \left( \frac{c}{k^3\log L}\right) ,\ \ \text { or }\ \ \mathsf{opt} (\mathcal {H}) \le O\left( (\log L)^k\right) \cdot s,$$

assuming the Unique Games Conjecture.

Subhash Khot—Research supported by NSF grants CCF 1422159, 1061938, 0832795 and Simons Collaboration on Algorithms and Geometry grant.