Vector Balancing in Polynomial Time

2026-09-20

AI Summary

本文提出了一种谱签名算法，通过最小化三次谱势，以多项式时间解决Komlós问题，找到满足特定条件的向量。

Abstract

We present a spectral signing algorithm solving the Komlós problem with a constant discrepancy in polynomial time. Given a matrix $A\in\mathbb{R}^{m\times n}$ whose columns have Euclidean norm at most $1$, the algorithm finds a vector $\varepsilon\in\{-1,1\}^n$ satisfying $\|A\varepsilon\|_\infty\le C$, where $C$ is an absolute constant. By minimizing a cubic spectral potential, our spectral signing algorithm updates the fractional coloring toward Boolean signs with time complexity $O((mn^9+n^{10})\log(2+m+n))$.

Problem

• vector balancing
• Komlós problem
• constant discrepancy

Innovation

• spectral signing algorithm
• polynomial time
• constant discrepancy
• cubic spectral potential