Minimum Cost Homomorphisms with Constrained Costs

Abstract

The minimum cost homomorphism problem is a natural optimization problem for homomorphisms to a fixed graph H. Given an input graph G, with a cost associated with mapping any vertex of G to any vertex of H, one seeks to minimize the sum of costs of the assignments over all homomorphisms of G to H. The complexity of this problem is well understood, as a function of the target graph H. For bipartite graphs H, the problem is polynomial time solvable if H is a proper interval bigraph, and is NP-complete otherwise. In many applications, the costs may be assumed to be the same for all vertices of the input graph. We study the complexity of this restricted version of the minimum cost homomorphism problem. Of course, the polynomial cases are still polynomial under this restriction. We expect the same will be true for the NP-complete cases, i.e., that the complexity classification will remain the same under the restriction. We verify this for the class of trees. For general graphs H, we prove a partial result: the problem is polynomial if H is a proper interval bigraph and is NP-complete when H is not chordal bipartite.