Polynomial-Time Algorithms for Weaver's Discrepancy Problem in a Dense Regime
<|reference_start|>Polynomial-Time Algorithms for Weaver's Discrepancy Problem in a Dense Regime: Given $v_1,\ldots, v_m\in\mathbb{C}^d$ with $\|v_i\|^2= \alpha$ for all $i\in[m]$ as input and suppose $\sum_{i=1}^m | \langle u, v_i \rangle |^2 = 1$ for every unit vector $u\in\mathbb{C}^d$, Weaver's discrepancy problem asks for a partition $S_1, S_2$ of $[m]$, such that $\sum_{i\in S_{j}} |\langle u, v_i \rangle|^2 \leq 1 -\theta$ for some universal constant $\theta$, every unit vector $u\in\mathbb{C}^d$ and every $j\in\{1,2\}$. We prove that this problem can be solved deterministically in polynomial time when $m\geq 49 d^2$.<|reference_end|>
@article{jourdan2024polynomial-time,
    title={Polynomial-Time Algorithms for Weaver's Discrepancy Problem in a Dense
  Regime},
    author={Ben Jourdan, Peter Macgregor, He Sun},
    journal={arXiv preprint arXiv:2402.08545},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.08545},
    primaryClass={cs.DS}
}