Hardness Results for Weaver's Discrepancy Problem
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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 √(8 α) + 2 α. 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 κ√(α), for some absolute constant κ > 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.
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