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 real-world 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 uV (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 ...

• Type: Journal
• Year: 2008
• Where: DAM
• Authors: Gregory Gutin, Arash Rafiey, Anders Yeo
• Keywords: DAM 2008, Homomorphism, Minimum Cost Homomorphism, Positive Integral Cost