License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.14
URN: urn:nbn:de:0030-drops-163556
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16355/
Go to the corresponding LIPIcs Volume Portal

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

Smoothed Analysis of the Komlós Conjecture

pdf-format:
LIPIcs-ICALP-2022-14.pdf (0.6 MB)

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 ?_∞ 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.

BibTeX - Entry

@InProceedings{bansal_et_al:LIPIcs.ICALP.2022.14,
  author =	{Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand},
  title =	{{Smoothed Analysis of the Koml\'{o}s Conjecture}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{14:1--14:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16355},
  URN =		{urn:nbn:de:0030-drops-163556},
  doi =		{10.4230/LIPIcs.ICALP.2022.14},
  annote =	{Keywords: Koml\'{o}s conjecture, smoothed analysis, weighted second moment method, subgaussian coloring}
}

Keywords: Komlós conjecture, smoothed analysis, weighted second moment method, subgaussian coloring
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI