Article

# Minimum Cost Homomorphism Dichotomy for Oriented Cycles.

Graphs and Combinatorics (Impact Factor: 0.39). 11/2009; 25(4):521-531. DOI: 10.1007/s00373-009-0853-9

Source: DBLP

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**ABSTRACT:**We introduce a class of digraphs analogous to proper interval graphs and bigraphs. They are defined via a geometric representation by two inclusion-free families of intervals satisfying a certain monotonicity condition; hence we call them monotone proper interval digraphs. They admit a number of equivalent definitions, including an ordering characterization by so-called Min-Max orderings, and the existence of certain graph polymorphisms. Min-Max orderings arose in the study of minimum cost homomorphism problems: if $H$ admits a a Min-Max ordering (or a certain extension of Min-Max orderings), then the minimum cost homomorphism problem to $H$ is known to admit a polynomial time algorithm. We give a forbidden structure characterization of monotone proper interval digraphs, which implies a polynomial time recognition algorithm. This characterizes digraphs with a Min-Max ordering; we also similarly characterize digraphs with an extended Min-Max ordering. In a companion paper, we shall apply this latter characterization to derive a conjectured dichotomy classification for the minimum cost homomorphism problems---namely, we shall prove that the minimum cost homomorphism problem to a digraph that does not admit an extended Min-Max ordering is NP-complete.SIAM Journal on Discrete Mathematics 01/2012; 26(4):1576-1596. DOI:10.1137/100783844 · 0.65 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**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.SIAM Journal on Discrete Mathematics 01/2012; 26(4):1597–1608. DOI:10.1137/100783856 · 0.65 Impact Factor