Document Open Access Logo

Smoothed Analysis of the Komlós Conjecture

Authors Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, Makrand Sinha

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2022.14.pdf
  • Filesize: 0.64 MB
  • 12 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Komlós conjecture
  • smoothed analysis
  • weighted second moment method
  • subgaussian coloring

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The well-known Komlós conjecture states that given n vectors in ℝ^d with Euclidean norm at most one, there always exists a ± 1 coloring such that the 𝓁_∞ norm of the signed-sum vector is a constant independent of n and d. We prove this conjecture in a smoothed analysis setting where the vectors are perturbed by adding a small Gaussian noise and when the number of vectors n = ω(d log d). The dependence of n on d is the best possible even in a completely random setting. 
Our proof relies on a weighted second moment method, where instead of considering uniformly randomly colorings we apply the second moment method on an implicit distribution on colorings obtained by applying the Gram-Schmidt walk algorithm to a suitable set of vectors. The main technical idea is to use various properties of these colorings, including subgaussianity, to control the second moment.

Cite As Get BibTex

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Smoothed Analysis of the Komlós Conjecture. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.14

Author Details

Nikhil Bansal
  • University of Michigan, Ann Arbor, MI, USA
Haotian Jiang
  • University of Washington, Seattle, WA, USA
Raghu Meka
  • University of California, Los Angeles, CA, USA
Sahil Singla
  • Georgia Institute of Technology, Atlanta, GA, USA
Makrand Sinha
  • Simons Institute and University of California, Berkeley, CA, USA

Funding

  • Bansal, Nikhil: Supported in part by the NWO VICI grant 639.023.812.
  • Jiang, Haotian: Supported by NSF awards CCF-1749609, DMS-1839116, DMS-2023166, CCF-2105772.
  • Sinha, Makrand: Supported by a Simons-Berkeley postdoctoral fellowship.

References

  1. Dimitris Achlioptas and Yuval Peres. The threshold for random k-sat is 2^klog 2-o(k). J. Amer. Math. Soc., 4:947-973, 2004. Google Scholar
  2. Noga Alon and Joel H. Spencer. The Probabilistic Method. John Wiley & Sons, 2016. Google Scholar
  3. Dylan J. Altschuler and Jonathan Niles-Weed. The discrepancy of random rectangular matrices. CoRR, abs/2101.04036, 2021. URL: http://arxiv.org/abs/2101.04036.
  4. Wojciech Banaszczyk. Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Struct. Algorithms, 12(4):351-360, 1998. Google Scholar
  5. Nikhil Bansal, Daniel Dadush, Shashwat Garg, and Shachar Lovett. The Gram-Schmidt walk: A cure for the Banaszczyk blues. Theory Comput., 15:1-27, 2019. Google Scholar
  6. Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Prefix discrepancy, smoothed analysis, and combinatorial vector balancing. In Proceedings of ITCS, pages 13:1-13:22, 2022. Google Scholar
  7. Nikhil Bansal and Raghu Meka. On the discrepancy of random low degree set systems. In Proceedings of SODA 2019, pages 2557-2564, 2019. Google Scholar
  8. József Beck and Tibor Fiala. "Integer-making" theorems. Discrete Appl. Math., 3(1):1-8, 1981. Google Scholar
  9. Esther Ezra and Shachar Lovett. On the Beck-Fiala conjecture for random set systems. Random Struct. Algorithms, 54(4):665-675, 2019. Google Scholar
  10. Cole Franks and Michael Saks. On the discrepancy of random matrices with many columns. Random Struct. Algorithms, 57(1):64-96, 2020. Google Scholar
  11. Nika Haghtalab, Tim Roughgarden, and Abhishek Shetty. Smoothed analysis with adaptive adversaries. In Proceedings of FOCS, 2021. Google Scholar
  12. Christopher Harshaw, Fredrik Sävje, Daniel Spielman, and Peng Zhang. Balancing covariates in randomized experiments using the gram-schmidt walk. arXiv e-prints, 2019. URL: http://arxiv.org/abs/1911.03071.
  13. Rebecca Hoberg and Thomas Rothvoss. A Fourier-Analytic Approach for the Discrepancy of Random Set Systems. In Proceedings of SODA, pages 2547-2556, 2019. Google Scholar
  14. Calum MacRury, Tomáš Masařík, Leilani Pai, and Xavier Pérez-Giménez. The phase transition of discrepancy in random hypergraphs, 2021. URL: http://arxiv.org/abs/2102.07342.
  15. Aditya Potukuchi. Discrepancy in random hypergraph models. CoRR, abs/1811.01491, 2018. Google Scholar
  16. Aditya Potukuchi. A spectral bound on hypergraph discrepancy. In Proceedings of ICALP, pages 93:1-93:14, 2020. Google Scholar
  17. Daniel A Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM (JACM), 51(3):385-463, 2004. Google Scholar
  18. Paxton Turner, Raghu Meka, and Philippe Rigollet. Balancing gaussian vectors in high dimension. In Proceedings of COLT, pages 3455-3486, 2020. Google Scholar
  19. Roman Vershynin. High-Dimensional Probability, volume 1. Cambridge University Press, 2018. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact