Minimum Cost and List Homomorphisms to Semicomplete Digraphs ## AbstractFor 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 $f(u)f(v)\in A(H).$ Let $H$ be a fixed directed or undirected graph. The homomorphism problem for $H$ asks whether a directed or undirected input graph $D$ admits a homomorphism to $H.$ The list homomorphism problem for $H$ is a generalization of the homomorphism problem for $H$, where every vertex $x\in V(D)$ is assigned a set $L_x$ of possible colors (vertices of $H$). The following optimization version of these decision problems 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.
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