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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 unanimitypolymorphism $\phi$ then deciding whether $G$ admits a homomorphism to $H$(HOM($H$)) is polynomial-time solvable. This gives proof of the dichotomyconjecture (now dichotomy theorem) by Feder and Vardi. Our approach iscombinatorial, and it is simpler than the two algorithms found by Bulatov andZhuk. We have implemented our algorithm and show some experimental results. Weuse our algorithm together with the recent result [38] for recognition ofMaltsev polymorphisms and decide in polynomial time if a given relationalstructure $\mathcal{R}$ admits a weak near unanimity polymorphism.

Digraph homomorphism problem and weak near unanimity polymorphism