🤖 AI Summary
This paper addresses the low-coloring problem for linearly ordered (LO) hypergraphs: given an LO 2-colorable 3-uniform hypergraph, how to construct an LO *k*-coloring with minimal *k* in polynomial time. Prior state-of-the-art algorithms achieved only an *O*(n^{1/5})-coloring—a polynomial upper bound. We present the first simple polynomial-time algorithm for this problem, based on combinatorial construction and a greedy layering strategy, without requiring semidefinite programming (SDP). Our algorithm outputs an LO *O*(log₂ n)-coloring on any LO 2-colorable 3-uniform hypergraph. This improves the best-known upper bound on the LO chromatic number from polynomial to logarithmic—yielding an exponential improvement—and constitutes the first logarithmic approximation algorithm for LO hypergraph coloring.
📝 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 $kgeq 2$ [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.