Hardness Results for Weaver’s Discrepancy Problem
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.
Discrepancy Problem
Kadison-Singer Problem
Hardness of Approximation
Theory of computation~Problems, reductions and completeness
Theory of computation~Design and analysis of algorithms
40:1-40:14
APPROX
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.
Daniel A.
Spielman
Daniel A. Spielman
Yale University, New Haven, CT, USA
http://www.cs.yale.edu/homes/spielman/
Peng
Zhang
Peng Zhang
Rutgers University, Piscataway, NJ, USA
https://sites.google.com/site/pengzhang27182/
10.4230/LIPIcs.APPROX/RANDOM.2022.40
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Daniel A. Spielman and Peng Zhang
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