Extensions of the Minimum Cost Homomorphism Problem

TL;DR

This paper extends the dichotomy classification of the minimum cost homomorphism problem to a broader class with conservative constraint languages, establishing polynomial-time solvability or NP-completeness under certain conditions.

Contribution

It generalizes the dichotomy theorem for MinHom to conservative constraint languages and provides new complexity classifications for these problems.

Findings

Dichotomy established for conservative MinHom with certain restrictions on R.

Polynomial-time solvability for specific conservative constraint languages.

NP-completeness results for other classes within the conservative framework.

Abstract

Assume $D$ is a finite set and $R$ is a finite set of functions from $D$ to the natural numbers. An instance of the minimum $R$-cost homomorphism problem ($MinHom_R$) is a set of variables $V$ subject to specified constraints together with a positive weight $c_{vr}$ for each combination of $v \in V$ and $r \in R$. The aim is to find a function $f:V \rightarrow D$ such that $f$ satisfies all constraints and $\sum_{v \in V} \sum_{r \in R} c_{vr}r(f(v))$ is minimized. This problem unifies well-known optimization problems such as the minimum cost homomorphism problem and the maximum solution problem, and this makes it a computationally interesting fragment of the valued CSP framework for optimization problems. We parameterize $MinHom_R\left(\Gamma\right)$ by {\em constraint languages}, i.e. sets $\Gamma$ of relations that are allowed in constraints. A constraint language is called {\em conservative} if every unary relation is a member of it; such constraint languages play an important role in understanding the structure of constraint problems. The dichotomy conjecture for $MinHom_R$ is the following statement: if $\Gamma$ is a constraint language, then $MinHom_R\left(\Gamma\right)$ is either polynomial-time solvable or NP-complete. For $MinHom$ the dichotomy result has been recently obtained [Takhanov, STACS, 2010] and the goal of this paper is to expand this result to the case of $MinHom_R$ with conservative constraint language. For arbitrary $R$ this problem is still open, but assuming certain restrictions on $R$ we prove a dichotomy. As a consequence of this result we obtain a dichotomy for the conservative maximum solution problem.