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 uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is ∑ u∈V (G) cf(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 ci(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. 1 Motivation and Terminology We consider finite graphs (and digraphs) without multiple edges, but with loops allowed. For a graph (or digraph) H, we use V (H) and E(H)) to ∗Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, [email protected] and Department of Computer Science, University of Haifa, Israel †School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A 1S6, [email protected] ‡Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, [email protected] §Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, [email protected]