Communication: Minimum cost homomorphisms to semicomplete multipartite digraphs

  • Authors:

  • Gregory Gutin;Arash Rafiey;Anders Yeo

  • Affiliations:

  • Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK and Department of Computer Science, University of Haifa, Israel;School of Computing, Simon Fraser University, Burnaby, BC, V5A 1S6 Canada;Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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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 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.