Minimum Cost Homomorphisms with Constrained Costs

TL;DR

This paper investigates minimum cost homomorphism problems with constrained costs, establishing complexity classifications for trees and partial results for general graphs, linking problem complexity to graph properties.

Contribution

It provides a dichotomy classification for trees and partial results for general graphs regarding the complexity of constrained cost homomorphism problems.

Findings

Dichotomy for trees matches standard homomorphism problems.

Polynomial cases when H is a proper interval graph.

NP-complete cases when H is not chordal bipartite.

Abstract

Minimum cost homomorphism problems can be viewed as a generalization of list homomorphism problems. They also extend two well-known graph colouring problems: the minimum colour sum problem and the optimum cost chromatic partition problem. In both of these problems, the cost function meets an additional constraint: the cost of using a specific colour is the same for every vertex of the input graph. We study minimum cost homomorphism problems with cost functions constrained to have this property. Clearly, when the standard minimum cost homomorphism problem is polynomial, then the problem with constrained costs is also polynomial. We expect that the same may hold for the cases when the standard minimum cost homomorphism problem is NP-complete. We prove that this is the case for trees $H$: we obtain a dichotomy of minimum constrained cost homomorphism problems which coincides with the dichotomy of standard minimum cost homomorphism problems. For general graphs $H$, we prove a partial dichotomy: the problem is polynomial if $H$ is a proper interval graph and NP-complete when $H$ is not chordal bipartite.