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Minimum Cost Homomorphisms to
Semicomplete Bipartite Digraphs
G Gutin, A Rafiey and A Yeo
SIAM J. Discrete Mathematics
2006.
AbstractFor digraphs $D$ and $H$, a mapping $f:\ V(D)\dom V(H)$ is a
homomorphism of $G$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in
E(H).$ 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. [Edit]
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