Approximation Algorithm for Non-Boolean MAX k-CSP

TL;DR

This paper introduces a randomized polynomial-time approximation algorithm for non-Boolean MAX k-CSP problems, achieving near-optimal approximation ratios under the Unique Games Conjecture, and improves guarantees for the Boolean MAX k-CSP2 case.

Contribution

The paper presents a new approximation algorithm for k-CSPd with an asymptotically optimal ratio under the UGC, improving upon previous algorithms, and also enhances guarantees for Boolean MAX k-CSP2.

Findings

Approximation factor Omega(kd/d^k) for k-CSPd when k > Omega(log d)

Algorithm's bound is asymptotically optimal assuming the Unique Games Conjecture

Improved approximation guarantee for Boolean MAX k-CSP2

Abstract

In this paper, we present a randomized polynomial-time approximation algorithm for k-CSPd. In k-CSPd, we are given a set of predicates of arity k over an alphabet of size d. Our goal is to find an assignment that maximizes the number of satisfied constraints. Our algorithm has approximation factor Omega(kd/d^k) (when k > \Omega(log d)). This bound is asymptotically optimal assuming the Unique Games Conjecture. The best previously known algorithm has approximation factor Omega(k log d/d^k). We also give an approximation algorithm for the boolean MAX k-CSP2 problem with a slightly improved approximation guarantee.