Digraph homomorphism problem and weak near unanimity polymorphism

TL;DR

This paper proves that the digraph homomorphism problem is polynomial-time solvable when the target digraph admits a weak near unanimity polymorphism, confirming the Feder-Vardi dichotomy conjecture with a simpler combinatorial approach.

Contribution

It establishes a polynomial-time algorithm for the homomorphism problem under weak near unanimity polymorphisms, simplifying previous methods and confirming the dichotomy theorem.

Findings

Polynomial-time algorithm for digraph homomorphism with weak near unanimity polymorphisms

Confirmation of the Feder-Vardi dichotomy conjecture

Experimental validation of the proposed algorithm

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