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Improved Linearly Ordered Colorings of Hypergraphs via SDP Rounding

Authors Anand Louis , Alantha Newman, Arka Ray

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Author Details

Anand Louis
  • Indian Institute of Science, Bengaluru, India
Alantha Newman
  • Université Grenoble Alpes, France
Arka Ray
  • Indian Institute of Science, Bengaluru, India

Funding

  • Louis, Anand: Supported in part by SERB Award CRG/2023/002896 and the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).
  • Ray, Arka: Supported in part by the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).

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Anand Louis, Alantha Newman, and Arka Ray. Improved Linearly Ordered Colorings of Hypergraphs via SDP Rounding. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.30

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 Õ(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 Õ(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 in this case to obtain improved bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • hypergraph coloring
  • SDP rounding
  • promise constraint satisfaction problems

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