  For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a homomorphism of
$D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ If, moreover, each
vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the
cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed
digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$. The
problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D),
i\in V(H)$, whether there exists a homomorphism of $D$ 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
describe a dichotomy of the minimum cost homomorphism problem for semicomplete
multipartite digraphs $H$. This solves an open problem from an earlier paper.
To obtain the dichotomy of this paper, we introduce and study a new notion, a
$k$-Min-Max ordering of digraphs.
