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Hardness Results for Weaver’s Discrepancy Problem

Authors Daniel A. Spielman, Peng Zhang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Discrepancy Problem
  • Kadison-Singer Problem
  • Hardness of Approximation

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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.

Cite As Get BibTex

Daniel A. Spielman and Peng Zhang. Hardness Results for Weaver’s Discrepancy Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.40

Author Details

Daniel A. Spielman
  • Yale University, New Haven, CT, USA
Peng Zhang
  • Rutgers University, Piscataway, NJ, USA

Funding

This work was supported in part by NSF Grant CCF-1562041, ONR Award N00014-20-1-2335, and a Simons Investigator Award to Daniel Spielman.

References

  1. Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Nikhil Srivastava. Approximating the largest root and applications to interlacing families. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1015-1028. SIAM, 2018. Google Scholar
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM (JACM), 45(3):501-555, 1998. Google Scholar
  3. Nikhil Bansal. Constructive algorithms for discrepancy minimization. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 3-10. IEEE, 2010. Google Scholar
  4. Marcin Bownik, Pete Casazza, Adam W Marcus, and Darrin Speegle. Improved bounds in weaver and feichtinger conjectures. Journal für die reine und angewandte Mathematik (Crelles Journal), 2019(749):267-293, 2019. Google Scholar
  5. Petter Brändén. Hyperbolic polynomials and the Kadison-Singer problem. arXiv preprint, 2018. URL: http://arxiv.org/abs/1809.03255.
  6. Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. Clustering with qualitative information. Journal of Computer and System Sciences, 71(3):360-383, 2005. Google Scholar
  7. Moses Charikar, Alantha Newman, and Aleksandar Nikolov. Tight hardness results for minimizing discrepancy. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1607-1614. SIAM, 2011. Google Scholar
  8. Michael B. Cohen. Improved spectral sparsification and Kadison-Singer for sums of higher-rank matrices, 2016. URL: http://www.birs.ca/events/2016/5-day-workshops/16w5111/videos/watch/201608011534-Cohen.html.
  9. Venkatesan Guruswami. Inapproximability results for set splitting and satisfiability problems with no mixed clauses. Algorithmica, 38(3):451-469, 2004. Google Scholar
  10. Johan Håstad. Some optimal inapproximability results. Journal of the ACM (JACM), 48(4):798-859, 2001. Google Scholar
  11. Ben Jourdan, Peter Macgregor, and He Sun. Is the algorithmic kadison-singer problem hard? arXiv preprint, 2022. URL: http://arxiv.org/abs/2205.02161.
  12. Rasmus Kyng, Kyle Luh, and Zhao Song. Four deviations suffice for rank 1 matrices. Advances in Mathematics, 375:107366, 2020. URL: https://doi.org/10.1016/j.aim.2020.107366.
  13. A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. Google Scholar
  14. Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families ii: Mixed characteristic polynomials and the kadison—singer problem. Annals of Mathematics, pages 327-350, 2015. Google Scholar
  15. G. A. Margulis. Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24(1):39-46, July 1988. Google Scholar
  16. Joel Spencer. Six standard deviations suffice. Transactions of the American mathematical society, 289(2):679-706, 1985. Google Scholar
  17. Nik Weaver. The Kadison-Singer problem in discrepancy theory. Discrete mathematics, 278(1-3):227-239, 2004. Google Scholar
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