THE DICHOTOMY OF MINIMUM COST HOMOMORPHISM PROBLEMS FOR DIGRAPHS

The minimum cost homomorphism problem has arisen as a natural and useful optimization problem in the study of graph (and digraph) coloring and homomorphisms: it unifies a number of other well studied optimization problems. It was shown by Gutin, Rafiey, and Yeo that the minimum cost problem for homomorphisms to a digraph H that admits a so-called extended Min-Max ordering is polynomial time solvable, and these authors conjectured that for all other digraphs H the problem is NP-complete. In a companion paper, we gave a forbidden structure characterization of digraphs that admit extended Min-Max orderings. In this paper, we apply this characterization to prove Gutin's conjecture.