Technology Category
Constraint Satisfaction and Optimization: Distributed CSP/OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Graphical Models
The sorrows of a smooth digraph: the first hardness criterion for infinite directed graph-colouring problems
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Infinite Directed Graphs
Coloring Problem
Complexity Classification
Innovation
Methods, ideas, or system contributions that make the work stand out.
Directed Graph Classification
Infinite Structures
NP-hard Problem
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