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

TL;DR

This work extends finite constraint satisfaction dichotomies to infinite ω-categorical structures by lifting techniques via an α-refinement framework and a 2-conservative Ω-orbit perspective. The authors prove that for a smooth digraph G of algebraic length 1 with oligomorphic automorphisms, either G/Ω contains a pseudoloop or the expansion by Ω-orbits pp-constructs every finite structure, implying NP-hardness for the related conservative colouring problem in the absence of a pseudoloop. A central innovation is the finitising equivalence α( G, Ω ), which yields finite quotients on components and enables a master induction that uses reductionistic sets to manage the infinite structure. The paper further derives a 4-ary pseudo-Siggers polymorphism in the non-hard case and introduces an algebraic invariant for ω-categorical structures enriched by pairs of orbits, substantially advancing the understanding of lifting finite CSP results to the infinite, highly symmetric setting.

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 $ω$-categorical structures; the strongest lifting results hitherto not going beyond a generalisation of the Hell-Nešetřil theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary $ω$-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.

Figures (2)

• Figure 1: $\omega$ vs. $\alpha$
• Figure 2: Overview of the proof

Theorems & Definitions (104)

• Example 1.1
• Example 1.2
• Theorem 1.2
• Corollary 1.3
• Example 1.4
• Corollary 1.4
• Proposition 1.4
• Corollary 3.0
• proof : Proof of \ref{['cor:siggers']} from \ref{['thm:10']}
• Lemma 4.0