The 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 problemwasmotivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. 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 theminimum cost homomorphism problem for H and denote it asMinHOM(H). The problem is to decide, for an input graph Dwith 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 Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12,11]. 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 Gutin and Kim (2008) [9]. © 2009 Elsevier B.V. All rights reserved.