Approximation of non-boolean 2CSP

TL;DR

This paper presents a polynomial-time algorithm using semidefinite programming to approximate Max 2CSP-R problems, achieving near-optimal results under the Unique Games Conjecture, improving over previous LP-based methods.

Contribution

The paper introduces a novel SDP-based approximation algorithm with a new rounding technique for Max 2CSP-R, matching the conjectured optimal approximation ratio.

Findings

Achieves an $rac{1}{R} ext{log} R$ approximation ratio.

Provides an SDP with an almost-matching integrality gap.

Improves upon the previous $1/R$-approximate linear programming approach.

Abstract

We develop a polynomial time $\Omega\left ( \frac 1R \log R \right)$ approximate algorithm for Max 2CSP-$R$, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size $R$, and we want to find an assignment to the variables that maximizes the number of satisfied constraints. Assuming the Unique Games Conjecture, this is the best possible approximation up to constant factors. Previously, a $1/R$-approximate algorithm was known, based on linear programming. Our algorithm is based on semidefinite programming (SDP) and on a novel rounding technique. The SDP that we use has an almost-matching integrality gap.