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URN: urn:nbn:de:0030-drops-125002
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A Spectral Bound on Hypergraph Discrepancy

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Abstract

Let ℋ be a t-regular hypergraph on n vertices and m edges. Let M be the m × n incidence matrix of ℋ and let us denote λ = max_{v ∈ 𝟏^⟂} 1/‖v‖ ‖Mv‖. We show that the discrepancy of ℋ is O(√t + λ). As a corollary, this gives us that for every t, the discrepancy of a random t-regular hypergraph with n vertices and m ≥ n edges is almost surely O(√t) as n grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.

BibTeX - Entry

@InProceedings{potukuchi:LIPIcs:2020:12500,
  author =	{Aditya Potukuchi},
  title =	{{A Spectral Bound on Hypergraph Discrepancy}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{93:1--93:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12500},
  URN =		{urn:nbn:de:0030-drops-125002},
  doi =		{10.4230/LIPIcs.ICALP.2020.93},
  annote =	{Keywords: Hypergraph discrepancy, Spectral methods, Beck-Fiala conjecture}
}

Keywords: Hypergraph discrepancy, Spectral methods, Beck-Fiala conjecture
Seminar: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue date: 2020
Date of publication: 29.06.2020


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