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 graph input digraph 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 Lx of possible colors (vertices of H). The following optimization version of these decision problems was introduced in [16], 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 ci(u) for each u ∈ V (D) and i ∈ V (H). The cost of a homomorphism f of D to H is u∈V (D) cf(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 ci(u) for each u ∈ V (D) and i ∈ 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 MinHOMP(H) 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. ∗Corresponding author. Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, [email protected] and Department of Computer Science, University of Haifa, Israel †Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, [email protected] ‡Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK, [email protected]