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A Spectral Bound on Hypergraph Discrepancy

Author Aditya Potukuchi

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Author Details

Aditya Potukuchi
  • Department of Computer Science, Rutgers University, New Brunswick, NJ, USA

Funding

  • Potukuchi, Aditya: Research supported in part by NSF grants CCF-1540634 and CCF-1814409.

Acknowledgements

I am extremely thankful to Jeff Kahn for suggesting the problem and to Shachar Lovett for the discussions that led to a refinement over a previous version. I would also like to thank Huseyin Acan and Cole Franks for the very helpful discussions. I would also like to thank the anonymous ICALP referees for the numerous detailed comments and suggestions.

Cite As Get BibTex

Aditya Potukuchi. A Spectral Bound on Hypergraph Discrepancy. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 93:1-93:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.93

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Hypergraph discrepancy
  • Spectral methods
  • Beck-Fiala conjecture

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