Towards a Characterization of Constant-Factor Approximable Finite-Valued CSPs

TL;DR

This paper characterizes when finite-valued constraint satisfaction problems can be approximated within a constant factor using algebraic conditions, providing new insights into their approximability and hardness.

Contribution

It introduces algebraic conditions that determine the finiteness of the LP relaxation gap for VCSPs and connects these conditions to approximation algorithms and hardness results.

Findings

Algebraic conditions for LP integrality gap finiteness

Efficient constant-factor approximation algorithms for certain VCSPs

NP-hardness results when algebraic conditions are not met

Abstract

In this paper we study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language {\Gamma} consisting of finitary functions on a fixed finite domain. An instance of VCSP is given by a finite set of variables and a sum of functions belonging to {\Gamma} and depending on a subset of the variables. Each function takes values in [0, 1] specifying costs of assignments of labels to its variables, and the goal is to find an assignment of labels to the variables that minimizes the sum. A recent result of Ene et al. says that, under the mild technical condition that {\Gamma} contains the equality relation, the basic LP relaxation is optimal for constant-factor approximation for VCSP({\Gamma}) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give new natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP({\Gamma}). We also show how these algebraic conditions can in principle be used to round solutions of the basic LP relaxation, and how, for several examples that cover all previously known cases, this leads to efficient constant-factor approximation algorithms. Finally, we show that the absence of another algebraic condition leads to NP-hardness of constant-factor approximation.