On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems

TL;DR

This paper investigates the approximability of minimum cost constraint satisfaction problems using algebraic properties, establishing a dichotomy based on the presence of certain polymorphisms and their implications for approximation algorithms.

Contribution

It introduces a new approximation algorithm for MinCostCSP with dual discriminator polymorphisms and characterizes when constant-factor approximations are possible based on NU polymorphisms.

Findings

Existence of a |D|-approximation algorithm for certain constraint languages.

Constant-factor approximability linked to the presence of NU polymorphisms.

NP-hardness of approximation for languages lacking NU polymorphisms, assuming UGC.

Abstract

We study minimum cost constraint satisfaction problems (MinCostCSP) through the algebraic lens. We show that for any constraint language $\Gamma$ which has the dual discriminator operation as a polymorphism, there exists a $|D|$-approximation algorithm for MinCostCSP$(\Gamma)$ where $D$ is the domain. Complementing our algorithmic result, we show that any constraint language $\Gamma$ where MinCostCSP$(\Gamma)$ admits a constant-factor approximation must have a \emph{near-unanimity} (NU) polymorphism unless P = NP, extending a similar result by Dalmau et al. on MinCSPs. These results imply a dichotomy of constant-factor approximability for constraint languages that contain all permutation relations (a natural generalization for Boolean CSPs that allow variable negation): either MinCostCSP$(\Gamma)$ has an NU polymorphism and is $|D|$-approximable, or it does not have any NU polymorphism and is NP-hard to approximate within any constant factor. Finally, we present a constraint language which has a majority polymorphism, but is nonetheless NP-hard to approximate within any constant factor assuming the Unique Games Conjecture, showing that the condition of having an NU polymorphism is in general not sufficient unless UGC fails.