Minimum cost homomorphism dichotomy for oriented cycles

For digraphs D and H, a mapping f : V(D) -> V(H) is a homomorphism of D to H if uv epsilon A(D) implies f(u) f(v) epsilon A(H). If, moreover, each vertex u epsilon V(D) is associated with costs c(i)(u), i epsilon V(H), then the cost of the homomorphism f is Sigma C-u epsilon V(D)(f(u))(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H (abbreviated MinHOM(H)). In this discrete optimization problem, we are to decide, for an input graph D with costs c(i)(u), u epsilon V(D),i epsilon V(H), whether there exists a homornorphism of D to H and, if one exists, to find one of minimum cost. We obtain a dichotomy classification for the time complexity of MinHOM(H) when H is an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops.