A logarithmic approximation of linearly-ordered colourings
(2404.19556)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. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement.
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