A dichotomy for minimum cost graph homomorphisms

For graphs G and H , a mapping f : V ( G ) → V ( H ) is a homomorphism of G to H if u v ∈ E ( G ) implies f ( u ) f ( v ) ∈ E ( H ) . If, moreover, each vertex u ∈ V ( G ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( G ) c f ( u ) ( u ) . For each fixed graph H , we have the minimum cost homomorphism problem, written as MinHOM ( H ) . The problem is to decide, for an input graph G with costs c i ( u ) , u ∈ V ( G ) , i ∈ V ( H ) , whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H , with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM ( H ) is polynomial time solvable. In all other cases the problem MinHOM ( H ) is NP-hard. This solves an open problem from an earlier paper.