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.
@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/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.40}, 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} }
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