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Smoothed Analysis of the Komlós Conjecture

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



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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

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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

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.

Subject Classification

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

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