Three-Element Min-Sol and Conservative Min-Cost-Hom

TL;DR

This paper classifies the computational complexity of two VCSP variants, Min-Cost-Hom and Min-Sol, on small and arbitrary domains, advancing understanding beyond finite-valued constraint languages.

Contribution

It provides a complete complexity classification for Min-Sol on domains up to three elements and for conservative Min-Cost-Hom on any finite domain.

Findings

Full classification for Min-Sol on domains with up to three elements.

Complete complexity characterization for conservative Min-Cost-Hom on arbitrary finite domains.

Answers a previously open question by Takhanov.

Abstract

Thapper and Zivny [STOC'13] recently classified the complexity of VCSP for all finite-valued constraint languages. However, the complexity of VCSPs for constraint languages that are not finite-valued remains poorly understood. In this paper we study the complexity of two such VCSPs, namely Min-Cost-Hom and Min-Sol. We obtain a full classification for the complexity of Min-Sol on domains that contain at most three elements and for the complexity of conservative Min-Cost-Hom on arbitrary finite domains. Our results answer a question raised by Takhanov [STACS'10, COCOON'10].