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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 AsGet 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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References

  1. Noga Alon and Joel H. Spencer. The probabilistic method, 2000. Google Scholar
  2. Nikhil Bansal. Constructive algorithms for discrepancy minimization. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 3-10, 2010. URL: https://doi.org/10.1109/FOCS.2010.7.
  3. Nikhil Bansal and Raghu Meka. On the discrepancy of random low degree set systems. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2557-2564, 2019. URL: https://doi.org/10.1137/1.9781611975482.157.
  4. József Beck. Roth’s estimate of the discrepancy of integer sequences is nearly sharp. Combinatorica, 1(4):319-325, 1981. URL: https://doi.org/10.1007/BF02579452.
  5. József Beck and Tibor Fiala. "integer-making" theorems. Discrete Applied Mathematics, 3(1):1-8, 1981. URL: https://doi.org/10.1016/0166-218X(81)90022-6.
  6. Debe Bednarchak and Martin Helm. A note on the beck-fiala theorem. Combinatorica, 17(1):147-149, March 1997. URL: https://doi.org/10.1007/BF01196138.
  7. Andrei Z. Broder, Alan M. Frieze, Stephen Suen, and Eli Upfal. Optimal construction of edge-disjoint paths in random graphs. SIAM J. Comput., 28(2):541-573, 1998. URL: https://doi.org/10.1137/S0097539795290805.
  8. Boris Bukh. An improvement of the beck-fiala theorem. Combinatorics, Probability and Computing, 25(3):380?398, 2016. URL: https://doi.org/10.1017/S0963548315000140.
  9. Bernard Chazelle. The Discrepancy Method: Randomness and Complexity. Cambridge University Press, New York, NY, USA, 2000. Google Scholar
  10. Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey. Internet Math., 3(1):79-127, 2006. URL: https://projecteuclid.org:443/euclid.im/1175266369.
  11. Esther Ezra and Shachar Lovett. On the beck-fiala conjecture for random set systems. In APPROX-RANDOM, 2015. Google Scholar
  12. C. Franks and M. Saks. On the Discrepancy of Random Matrices with Many Columns. ArXiv e-prints, July 2018. URL: http://arxiv.org/abs/1807.04318.
  13. J. Friedman, J. Kahn, and E. Szemerédi. On the second eigenvalue of random regular graphs. In Proceedings of the Twenty-first Annual ACM Symposium on Theory of Computing, STOC '89, pages 587-598, New York, NY, USA, 1989. ACM. URL: https://doi.org/10.1145/73007.73063.
  14. Rebecca Hoberg and Thomas Rothvoss. A fourier-analytic approach for the discrepancy of random set systems. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2547-2556. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.156.
  15. Greg Kuperberg, Shachar Lovett, and Ron Peled. Probabilistic existence of rigid combinatorial structures. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 1091-1106. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214075.
  16. Avi Levy, Harishchandra Ramadas, and Thomas Rothvoss. Deterministic discrepancy minimization via the multiplicative weight update method. In Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, pages 380-391, 2017. URL: https://doi.org/10.1007/978-3-319-59250-3_31.
  17. Shachar Lovett and Raghu Meka. Constructive discrepancy minimization by walking on the edges. SIAM J. Comput., 44(5):1573-1582, 2015. URL: https://doi.org/10.1137/130929400.
  18. J. Matousek. Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics. Springer Berlin Heidelberg, 1999. URL: https://books.google.com/books?id=BKvXj1GisP0C.
  19. Jiří Matoušek. Tight upper bounds for the discrepancy of half-spaces. Discrete & Computational Geometry, 13:593-601, 1995. URL: https://doi.org/10.1007/BF02574066.
  20. Aditya Potukuchi. Discrepancy in random hypergraph models, 2018. URL: http://arxiv.org/abs/1811.01491.
  21. Thomas Rothvoss. Constructive discrepancy minimization for convex sets. SIAM J. Comput., 46(1):224-234, 2017. URL: https://doi.org/10.1137/141000282.
  22. Joel Spencer. Six standard deviations suffice. Transactions of the American Mathematical Society, 289(2):679-706, 1985. URL: https://doi.org/10.1090/S0002-9947-1985-0784009-0.
  23. Joel Spencer. Coloring the projective plane. Discrete Mathematics, 73(1):213-220, 1988. URL: https://doi.org/10.1016/0012-365X(88)90150-1.
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