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

2025-01-28

AI Summary

This study addresses the conservative coloring problem for smooth directed graphs of algebraic length one that are ω-categorical, and classifies the computational complexity of associated constraint satisfaction problems (CSPs). For such graphs without pseudoloops, we establish the first algebraic criterion for NP-hardness: if no pseudoloop exists, there exists a pair of orbits whose primitive positive (pp) constructions generate all finite structures, rendering the conservative CSP NP-hard; otherwise, a polynomial-time algorithm exists. Methodologically, we extend the finite graph dichotomy theorem to the infinite ω-categorical setting, introducing a novel algebraic invariant—based on orbit pairs and pp-constructibility—previously applicable only to undirected graphs. By integrating model theory, universal algebra, oligomorphic group actions, and orbit analysis, we achieve a fundamental advance in characterizing the complexity of CSPs over infinite directed graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Social Networks and Social Media: Computational social science

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-Nev{s}etv{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.

Problem

• Infinite Directed Graphs
• Coloring Problem
• Complexity Classification

Innovation

• Directed Graph Classification
• Infinite Structures
• NP-hard Problem