Discrete Applied Mathematics
Minimum cost homomorphisms to oriented cycles with some loops
CATS '09 Proceedings of the Fifteenth Australasian Symposium on Computing: The Australasian Theory - Volume 94
Note: The Ck-extended graft construction
Discrete Applied Mathematics
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For digraphs $D$ and $H$, a mapping $f:V(D)\rightarrow V(H)$ 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 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 bipartite 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.