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Improved linearly ordered colorings of hypergraphs via SDP rounding

Anand Louis, Alantha Newman, Arka Ray

TL;DR

This work shows how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for hypergraphs, and shows how to reduce the problem to cases with highly structured SDP solutions, which are called balanced hypergraphs.

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 Živný 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.

Improved linearly ordered colorings of hypergraphs via SDP rounding

TL;DR

This work shows how to use SDP-based rounding methods to produce an LO coloring with colors for hypergraphs, and shows how to reduce the problem to cases with highly structured SDP solutions, which are called balanced hypergraphs.

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 Živný recently gave a polynomial-time algorithm to color such hypergraphs with 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 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.
Paper Structure (9 sections, 22 theorems, 45 equations, 2 algorithms)

This paper contains 9 sections, 22 theorems, 45 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Tools for LO Coloring and Proof of the Main Theorem
  3. Coloring by Finding Independent Sets
  4. Combinatorial Rounding for Unbalanced Vertices
  5. SDP Rounding for Balanced Hypergraphs
  6. Conclusion
  7. Properties of Gaussian
  8. Omitted Proofs
  9. Proof of lem:u_bound

Key Result

Theorem 1

Let $H$ be a 2-LO colorable 3-uniform hypergraph on $n$ vertices. Then there exists a (randomized) polynomial-time algorithm that finds an LO coloring of $H$ using $\widetilde{O}\left(n^{1/5} \right )$ colors.

Theorems & Definitions (43)

  • Theorem 1
  • Definition 2
  • Proposition 3: Proposition 3 in NZ22
  • Definition 5
  • Definition 6
  • Lemma 6
  • Proposition 7
  • proof
  • Corollary 8
  • Lemma 9
  • ...and 33 more