Complexity of the Minimum Cost Homomorphism Problem for Semicomplete Digraphs with Possible Loops

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). For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problem was motivated by a realworld problem in defence logistics and was introduced in [12]. If each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is ∑ u∈V (D) cf(u)(u). For each fixed digraph H , we have the minimum cost homomorphism problem for H and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph [9], and a semicomplete multipartite digraph [11, 10]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in [8].