Minimum cost and list homomorphisms to semicomplete digraphs

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism of D to H if uv∈A(D) implies f(u)f(v)∈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∈V(D) is assigned a set L_x of possible colors (vertices of H).The following optimization version of these decision problems generalizes the list homomorphism problem and was introduced in Gutin et al. Level of repair analysis and minimum cost homomorphisms of graphs, Discrete Appl. Math., to appear], 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 integral cost c_i(u) for each u∈V(D) and i∈V(H). The cost of a homomorphism f of D to H is . For a fixed digraph H, the minimum cost homomorphism problem for H is stated as follows: for an input digraph D and costs c_i(u) for each u∈V(D) and i∈V(H), verify whether there is a homomorphism of D to H and, if one exists, find such a homomorphism of minimum cost.We obtain dichotomy classifications of the computational complexity of the list homomorphism and minimum cost homomorphism problems, when H is a semicomplete digraph (digraph in which there is at least one arc between any two vertices). 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 the minimum cost homomorphism problem 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.