Dichotomy for Digraph Homomorphism Problems

TL;DR

This paper proves the dichotomy conjecture for digraph homomorphism problems by showing polynomial-time solvability when the fixed digraph admits a weak-near-unanimity polymorphism, using a simpler combinatorial approach.

Contribution

It provides a simpler combinatorial proof of the dichotomy theorem for digraph homomorphisms and includes an implementation with experimental results.

Findings

Deciding homomorphisms is polynomial time for digraphs with a weak-near-unanimity polymorphism.

The proof confirms the dichotomy conjecture for digraph homomorphism problems.

The approach simplifies previous algorithms by Bulatov and Zhuk.

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 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.