The Dichotomy of List Homomorphisms for Digraphs

TL;DR

This paper establishes a structural dichotomy for list homomorphism problems on digraphs, identifying a key graph property (DAT) that determines whether the problem is NP-complete or polynomial-time solvable.

Contribution

It introduces the concept of digraph asteroidal triples (DAT) and proves that their presence characterizes NP-completeness in list homomorphism problems for digraphs.

Findings

DAT-free digraphs have polynomial-time solvable problems

Presence of DAT implies NP-completeness

DAT can be recognized in polynomial time

Abstract

The Dichotomy Conjecture for constraint satisfaction problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction between the tractable and intractable cases. Such structural characterization is absent in Bulatov's classification, and Bulatov asked whether one can be found. We provide an answer in the case of digraphs; the technique will apply in a broader context. The key concept we introduce is that of a digraph asteroidal triple (DAT). The dichotomy then takes the following form. If a digraph H has a DAT, then the list homomorphism problem for H is NP-complete; and a DAT-free digraph H has a polynomial time solvable list homomorphism problem. DAT-free graphs can be recognized in polynomial time.