Dichotomy for Digraph Homomorphism Problems

We consider the problem of finding a homomorphism from an input digraph G to a fixed digraph H. We show that if H admits a weak-near-unanimity polymorphism φ then deciding whether G admits a homomorphism to H (HOM(H)) is polynomial time solvable. This confirms the conjecture of Bulatov, Jeavons, and Krokhin [BJK05], in the form postulated by Maroti and McKenzie [MM08], and consequently implies the validity of the celebrated dichotomy conjecture due to Feder and Vardi [FV93]. We transform the problem into an instance of the list homomorphism problem where initially all the lists are full (contain all the vertices of H). Then we use the polymorphism φ as a guide to reduce the lists to singleton lists, which yields a homomorphism if one exists.