We consider an optimization version of the homomorphism problem that was motivated by a real-world problem in defence logistics. For each vertex of input digraph D there is a cost (non-negative number) of mapping that vertex to a vertex of target digraph H. The minimum cost homomorphism problem, MinHOM(H), seeks a homomorphism f from D to H with minimum total cost. We give a classification of digraphs H for which MinHOM(H) is polynomial. Our classification is combinatorial and testable in polynomial time. The list homomorphism problem for H, LHOM(H), asks whether an input digraph D admits a homomorphism to H subject to restrictions (lists) of allowed images for each vertex of D. Clearly, LHOM(H) is reducible to MinHOM(H). The dichotomy for LHOM(H) was previously proved by Bulatov. If time allows I will also mention our combinatorial classification of digraphs H for which LHOM(H) problem is tractable. These are joint results with Pavol Hell.