Minimum cost homomorphisms to semicomplete multipartite digraphs |
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Authors: | Gregory Gutin Arash Rafiey |
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Institution: | a Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK b Department of Computer Science, University of Haifa, Israel c School of Computing, Simon Fraser University, Burnaby, BC, V5A 1S6 Canada |
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Abstract: | For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism ofDtoH 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 recently by the authors and M. Tso.Suppose we are given a pair of digraphs D,H and a 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 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. |
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Keywords: | Minimum cost homomorphism Semicomplete multipartite digraphs |
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