Minimum Cost Homomorphisms to Semicomplete Multipartite 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 integral 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). Let H be a fixed digraph. The minimum cost homomorphism problem for H , MinHOMP(H), is stated as follows: For input digraph D and costs ci(u) for each u ∈ V (D) and i ∈ V (H), verify whether there is a homomorphism of D to H and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of MinHOMP(H) for H being a semicomplete digraph. In this paper we extend the classification to semicomplete k-partite digraphs, k ≥ 3, and obtain such a classification for bipartite tournaments.