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.APPROX/RANDOM.2022.40
URN: urn:nbn:de:0030-drops-171628
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17162/
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Spielman, Daniel A. ; Zhang, Peng

Hardness Results for Weaver’s Discrepancy Problem

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LIPIcs-APPROX40.pdf (0.7 MB)

Abstract

Marcus, Spielman and Srivastava (Annals of Mathematics 2014) solved the Kadison-Singer Problem by proving a strong form of Weaver’s conjecture: they showed that for all α > 0 and all lists of vectors of norm at most √α whose outer products sum to the identity, there exists a signed sum of those outer products with operator norm at most √{8α} + 2α. We prove that it is NP-hard to distinguish such a list of vectors for which there is a signed sum that equals the zero matrix from those in which every signed sum has operator norm at least η √α, for some absolute constant η > 0. Thus, it is NP-hard to construct a signing that is a constant factor better than that guaranteed to exist.
For α = 1/4, we prove that it is NP-hard to distinguish whether there is a signed sum that equals the zero matrix from the case in which every signed sum has operator norm at least 1/4.

BibTeX - Entry

@InProceedings{spielman_et_al:LIPIcs.APPROX/RANDOM.2022.40,
  author =	{Spielman, Daniel A. and Zhang, Peng},
  title =	{{Hardness Results for Weaver’s Discrepancy Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{40:1--40:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17162},
  URN =		{urn:nbn:de:0030-drops-171628},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.40},
  annote =	{Keywords: Discrepancy Problem, Kadison-Singer Problem, Hardness of Approximation}
}

Keywords: Discrepancy Problem, Kadison-Singer Problem, Hardness of Approximation
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
Issue Date: 2022
Date of publication: 15.09.2022

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