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

## Abstract

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.