Minimum cost homomorphisms to locally semicomplete digraphs and quasi-transitive digraphs

A. Gupta, G. Gutin, M. Karimi, E. J. Kim, A. Rafiey

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Abstract

For digraphs G and H, a homomorphism of G to H is a mapping f : V(G)→V(H) such that uv ε A(G) implies f(u)f(v) ε A(H). If, moreover, each vertex u ε V(G) is associated with costs Ci(u), i ε V(H), then the cost of a homomorphism / is ∑uεv(G) cf(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), can be formulated as follows: Given an input digraph G, together with costs Ci(u), u ε V(G), i ε V(H), decide whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

Original languageEnglish
Pages (from-to)227-232
Number of pages6
JournalAustralasian Journal of Combinatorics
Volume46
Publication statusPublished - Feb 2010
Externally publishedYes

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ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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