eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-15
40:1
40:14
10.4230/LIPIcs.APPROX/RANDOM.2022.40
article
Hardness Results for Weaver’s Discrepancy Problem
Spielman, Daniel A.
1
Zhang, Peng
2
Yale University, New Haven, CT, USA
Rutgers University, Piscataway, NJ, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol245-approx-random2022/LIPIcs.APPROX-RANDOM.2022.40/LIPIcs.APPROX-RANDOM.2022.40.pdf
Discrepancy Problem
Kadison-Singer Problem
Hardness of Approximation