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 problem was motivated 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 c"i(u),i@?V(H), then the cost of the homomorphism f is @?"u"@?"V"("D")c"f"("u")(u). For each fixed digraph H, we have the minimum cost homomorphism problem forH and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs c"i(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].