Improved linearly ordered colorings of hypergraphs via SDP rounding

TL;DR

This paper introduces an SDP-based rounding technique to improve the number of colors needed for linearly ordered hypergraph coloring, reducing the bound from roughly n^{1/3} to n^{1/5}.

Contribution

It develops a novel SDP rounding approach for LO hypergraph coloring, achieving better bounds than previous polynomial-time algorithms.

Findings

LO coloring with n^{1/5} colors using SDP rounding

Reduction to balanced hypergraphs with structured SDP solutions

Application of classic SDP-rounding tools for improved bounds

Abstract

We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and \v{Z}ivn\'{y} recently gave a polynomial-time algorithm to color such hypergraphs with $\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We show how to reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then, we discuss how to apply classic SDP-rounding tools to obtain improved bounds.