Approximating CSPs Using LP Relaxation

Subhash Khot, Rishi Saket

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.