Weaver离散问题的困难性结果
Hardness Results for Weaver's Discrepancy Problem
摘要：
马库斯、斯皮尔曼和斯利瓦斯塔瓦（数学年鉴2014）通过证明韦弗猜想的强形式解决了Kadison-Singer问题：他们表明对于所有$α> 0$和所有至多$\sqrtα$的范数向量列表，其外积和为恒等式，存在一个算子范数至多为$\sqrt{8 α} + 2 α.$的外积的符号和，我们证明了它是NP-对于某个绝对常数$κ> 0.$，很难区分这样一个向量列表，其中存在等于零矩阵的符号和，以及其中每个符号和具有至少$κ\sqrtα$的算子范数。因此，构造一个比保证存在的常数因子更好的符号是NP困难的。对于$α= 1/4$，我们证明了区分是否存在等于零矩阵的符号和每个符号和至少有$1/4$的算子范数的情况是NP-困难的.
英文摘要：
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 $\sqrtα$ whose outer products sum to the identity, there exists a signed sum of those outer products with operator norm at most $\sqrt{8 α} + 2 α.$ 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 $κ\sqrtα$, 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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