Minimum Cost Homomorphism Dichotomy for Oriented Cycles
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 c i (u), i ∈ V(H), then the cost of the homomorphism f is ∑ u ∈ V(D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H (abbreviated MinHOM(H)). In this discrete optimization problem, we are to decide, for an input graph D with costs c i (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. We obtain a dichotomy classification for the time complexity of MinHOM(H) when H is an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops.
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- 1.Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer, London (2000)Google Scholar
- 3.Feder, T., Hell, P., Rafiey, A.: List homomorphism to balanced digraphs (submitted)Google Scholar
- 4.Gupta, A., Hell, P., Karimi, M., Rafiey, A.: Minimum cost homomorphisms to reflexive digraphs. In: LATIN 2008 (to appear, 2008)Google Scholar
- 5.Gutin, G., Hell, P., Rafiey, A., Yeo, A.: Minimum Cost Homomorphisms to Proper Interval Graphs and Bigraphs. Europ. J. Combin. (to appear)Google Scholar
- 7.Gutin, G., Rafiey, A., Yeo, A.: Minimum Cost Homomorphisms to Semicomplete Multipartite Digraphs. Discrete Applied Math. (to apear)Google Scholar
- 8.Gutin, G., Rafiey, A., Yeo, A.: Minimum Cost Homomorphisms to Semicomplete Bipartite Digraphs (submitted)Google Scholar
- 9.Gutin, G., Rafiey, A., Yeo, A., Tso, M.: Level of repair analysis and minimum cost homomorphisms of graphs. Discrete Appl. Math. 154, 881–889 (2006); Preliminary version appeared. In: Megiddo, N., Xu, Y., Zhu, B. (eds.): AAIM 2005. LNCS, vol. 3521, pp. 427–439. Springer, Heidelberg (2005)CrossRefMathSciNetzbMATHGoogle Scholar
- 10.Gutjahr, W.: Graph Colourings, PhD thesis, Free University, Berlin (1991)Google Scholar