Minimum Cost and List Homomorphisms to Semicomplete Digraphs
<|reference_start|>Minimum Cost and List Homomorphisms to Semicomplete Digraphs: The following optimization problem was introduced in \cite{gutinDAM}, where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs $D,H$ and a positive 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 $\sum_{u\in V(D)}c_{f(u)}(u)$. For a fixed digraph $H$, the minimum cost homomorphism problem for $H$, MinHOMP($H$), is stated as follows: For an 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 exists, find such a homomorphism of minimum cost.   We obtain dichotomy classifications of the computational complexity of the list homomorphism problem and MinHOMP($H$), when $H$ is a semicomplete digraph (a digraph in which every two vertices have at least one arc between them). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when $H$ is a semicomplete digraph: both problems are polynomial solvable if $H$ has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for \MiP is different: the problem is polynomial time solvable if $H$ is acyclic or $H$ is a cycle of length 2 or 3; otherwise, the problem is NP-hard.<|reference_end|>
@article{gutin2005minimum,
    title={Minimum Cost and List Homomorphisms to Semicomplete Digraphs},
    author={G. Gutin, A. Rafiey, A. Yeo},
    journal={arXiv preprint arXiv:cs/0507017},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0507017},
    primaryClass={cs.DM}
}