A logarithmic approximation of linearly ordered colourings

TL;DR

This paper introduces two simple polynomial-time algorithms that significantly improve the number of colours needed for linearly ordered colourings of hypergraphs, reducing it from polynomial bounds to logarithmic in the number of vertices.

Contribution

The paper presents the first polynomial-time algorithms achieving an $O(\log n)$ colour bound for LO colourings, an exponential improvement over previous bounds.

Findings

Algorithms achieve $O(\log n)$ colours for LO colourings.

Previous bounds were polynomial in $n$, now logarithmic.

The methods are simple and polynomial-time.

Abstract

A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is still open. Nakajima and \v{Z}ivn\'{y} gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with $O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$ colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with $O^*(\sqrt[5]{n})$ colours [FSTTCS'24]. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement.