PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Dichotomy for Minimum Cost Graph Homomorphisms
G. Gutin, P. Hell, A. Rafiey and A. Yeo
Europ. Journal Combinatorics Volume to appear, 2006.


For graphs $G$ and $H$, a mapping $f:\ V(G)\dom V(H)$ is a homomorphism of $G$ to $H$ if $uv\in E(G)$ implies $f(u)f(v)\in E(H).$ If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed graph $H$, we have the {\em minimum cost homomorphism problem}, written as MinHOM($H)$. The problem is to decide, for an input graph $G$ with costs $c_i(u),$ $u \in V(G), i\in V(H)$, 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. We prove a dichotomy of the minimum cost homomorphism problems for graphs $H$, with loops allowed. When each connected component of $H$ is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM($H)$ is polynomial time solvable. In all other cases the problem MinHOM($H)$ is NP-hard. This solves an open problem from an earlier paper.

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EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Theory & Algorithms
ID Code:2355
Deposited By:Gregory Gutin
Deposited On:22 November 2006