Dichotomy for Digraph Homomorphism Problems

Abstract

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 gives a proof of the dichotomy conjecture (now dichotomy theorem) by Feder and Vardi [29]. Our approach is combinatorial, and it is simpler than the two algorithms found by Bulatov [9] and Zhuk [46] in 2017. We have implemented our algorithm and show some experimental results.