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Hardness Results for Weaver’s Discrepancy Problem

Authors Daniel A. Spielman, Peng Zhang

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

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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)


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.

Subject Classification

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


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  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:
  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:
  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:
  12. Rasmus Kyng, Kyle Luh, and Zhao Song. Four deviations suffice for rank 1 matrices. Advances in Mathematics, 375:107366, 2020. URL:
  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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