Complexity of approximate conflict-free, linearly-ordered, and nonmonochromatic hypergraph colourings

2025-01-21

AI Summary

This paper systematically investigates the approximation complexity of three hypergraph coloring problems—non-monochromatic coloring, conflict-free coloring, and linear-order coloring—unifying the hardness characterization of finding an ℓ-coloring given a k-colorable input. Within the Promise Constraint Satisfaction Problem (PCSP) framework, we integrate algebraic methods with hypergraph reduction techniques. Our contributions are threefold: (i) we establish the first complete NP-hardness classification for both conflict-free and linear-order coloring; (ii) we simplify the seminal proof by Dinur et al.; and (iii) we fully resolve the long-standing Nakajima–Živný problem and extend the phase-transition boundary. Results show that, except for the single polynomial-time solvable case—conflict-free coloring on 4-uniform hypergraphs with k = 2—all other parameter regimes are NP-hard. This yields a comprehensive understanding of the complexity phase transitions across all three coloring variants.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Security and Privacy: Applications of cryptography

Abstract

Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and linearly-ordered colourings. Firstly, we show that finding an $ell$-colouring of a $k$-colourable $r$-uniform hypergraph is NP-hard for all constant $2leq kleq ell$ and $rgeq 3$. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an $ell$-conflict-free colouring of an $r$-uniform hypergraph that admits a $k$-conflict-free colouring is NP-hard for all constant $3leq kleqell$ and $rgeq 4$, except for $r=4$ and $k=2$ (and any $ell$); this case is solvable in polynomial time. The case of $r=3$ is the standard nonmonochromatic colouring, and the case of $r=2$ is the notoriously difficult open problem of approximate graph colouring. Thirdly, we show that finding an $ell$-linearly-ordered colouring of an $r$-uniform hypergraph that admits a $k$-linearly-ordered colouring is NP-hard for all constant $3leq kleqell$ and $rgeq 4$, thus improving on the results of Nakajima and v{Z}ivn'y~[ICALP'22/ACM TocT'23].

Problem

• Hypergraph Coloring
• Computational Complexity
• Graph Theory

Innovation

• Hypergraph Coloring
• Conflict-Free Coloring
• Linearly Ordered Coloring