  For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\
V(G)\dom V(H)$ such that $uv\in A(G)$ implies $f(u)f(v)\in A(H)$. If, moreover,
each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then
the cost of a homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. 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 $c_i(u)$, $u\in V(G)$, $i\in V(H)$, 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.

Discrete Mathematics