Minimum Cost Homomorphisms to Proper Interval Graphs and Bigraphs

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 describe 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. Along the way, we prove a new characterization of the class of proper interval bigraphs. 1 Motivation and Terminology We consider finite undirected and directed graphs without multiple edges, but with loops allowed. For a directed or undirected graph H, V (H) (E(H)) Corresponding author. 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]