The sorrows of a smooth digraph: the first hardness criterion for infinite directed graph-colouring problems

TL;DR

This paper extends complexity results from finite to infinite directed graphs, establishing a new hardness criterion for infinite directed graph-coloring problems involving smooth digraphs of algebraic length 1.

Contribution

It introduces the first hardness criterion for infinite directed graph-coloring problems, lifting finite structural results to the infinite setting using algebraic invariants.

Findings

Any smooth digraph of algebraic length 1 pp-constructs all finite structures unless it has a pseudo-loop.

The conservative graph-coloring problem is NP-hard under these conditions.

Established a new algebraic invariant for $$-categorical structures involving pairs of orbits.

Abstract

Two major milestones on the road to the full complexity dichotomy for finite-domain constraint satisfaction problems were Bulatov's proof of the dichotomy for conservative templates, and the structural dichotomy for smooth digraphs of algebraic length 1 due to Barto, Kozik, and Niven. We lift the combined scenario to the infinite, and prove that any smooth digraph of algebraic length 1 pp-constructs, together with pairs of orbits of an oligomorphic subgroup of its automorphism group, every finite structure -- and hence its conservative graph-colouring problem is NP-hard -- unless the digraph has a pseudo-loop, i.e. an edge within an orbit. We thereby overcome, for the first time, previous obstacles to lifting structural results for digraphs in this context from finite to $\omega$-categorical structures; the strongest lifting results hitherto not going beyond a generalisation of the Hell-Ne\v{s}et\v{r}il theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary $\omega$-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.