Minimum Cost Homomorphisms of Digraphs

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a realworld problem in defence logistics and was introduced very recently by the authors and M. Tso. Suppose we are given a pair of digraphs D,H and a positive cost ci(u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphism f of D to H is ∑ u∈V (D) cf(u)(u). For a fixed digraph H , the minimum cost homomorphism problem MinHOMP(H) is, for an input digraph D, to verify whether there is a homomorphism of D to H and, if it does exist, to find such a homomorphism of minimum cost. We introduce a general approach for showing that MinHOMP(H) is polynomial time solvable for some digraphs H . Using this method, we prove that MinHOMP(H) is polynomial time solvable when H is an acyclic multipartite tournament. This and other results allow us to obtain dichotomy classifications of computational complexity of MinHOMP(H) when H is a semicomplete m-partite, m ≥ 3, digraph or a bipartite tournament.