Introduction to the Minimum Cost Homomorphism Problem for Directed and Undirected Graphs

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) cf(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for 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. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems.