Minimum Cost Homomorphisms to Semicomplete Bipartite 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). If, moreover, each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is ∑ u∈V (D) cf(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H. The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D 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 problem for semicomplete multipartite digraphs H. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a k-Min-Max ordering of digraphs.