  For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a {\em homomorphism
of $D$ to $H$} if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ For a fixed directed
or undirected graph $H$ and an input graph $D$, the problem of verifying
whether there exists a homomorphism of $D$ to $H$ has been studied in a large
number of papers. We study an optimization version of this decision problem.
Our optimization problem is motivated by a real-world problem in defence
logistics and was introduced very recently by the authors and M. Tso.
  Suppose we are given a pair of digraphs $D,H$ and a positive integral cost
$c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$. The cost of a homomorphism $f$
of $D$ to $H$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. Let $H$ be a fixed digraph. The
minimum cost homomorphism problem for $H$, MinHOMP($H$), is stated as follows:
For input digraph $D$ and costs $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$,
verify whether there is a homomorphism of $D$ to $H$ and, if it does exist,
find such a homomorphism of minimum cost. In our previous paper we obtained a
dichotomy classification of the time complexity of \MiP for $H$ being a
semicomplete digraph. In this paper we extend the classification to
semicomplete $k$-partite digraphs, $k\ge 3$, and obtain such a classification
for bipartite tournaments.
