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 ci (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 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. 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.