Minimum cost homomorphisms to semicomplete multipartite digraphs

Abstract

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ 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 real-world problem in defence logistics and was introduced recently by the authors and M. Tso. Suppose we are given a pair of digraphs D , H and a cost c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) . The cost of a homomorphism f of D to H is ∑ u ∈ V ( D ) c f ( 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 c i ( 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 ) when H is 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.