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 ∈ A(D) implies f(u) f(v) ∈ A(H). If, moreover, 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 for H (abbreviated 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. 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.