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

TL;DR

The paper classifies the complexity of three hypergraph colouring variants—nonmonochromatic (NAE), conflict-free (CF), and linearly-ordered (LO)—within the algebraic PCSP framework. It provides a unified, simpler hardness proof for the NAE case using Kneser graphs and extends to CF and LO, showing $-\text{hard}$ness for broad parameter ranges, with explicit polynomial-time exceptions in the CF case (notably $(r,k)=(4,2)$). The methodology hinges on avoiding sets for polymorphisms and a chain-of-minors hardness criterion, supplemented by reductions via Kneser graphs and topological insights. These results advance unconditional hardness classifications for approximate hypergraph colourings and illuminate the landscape of PCSPs for colouring variants with practical implications for combinatorial optimization.

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 $2\leq k\leq \ell$ and $r\geq 3$. 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 $3\leq k\leq\ell$ and $r\geq 4$, 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 $3\leq k\leq\ell$ and $r\geq 4$, thus improving on the results of Nakajima and Živný [ICALP'22/ACM TocT'23].

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5: BG21:sicomp
• Lemma 6
• proof
• Theorem 7: BWZ21
• Theorem 8: Lovasz78
• Lemma 9