Near-Optimal Constructive Bounds for $\ell_2$ Prefix Discrepancy and Steinitz Problems via Affine Spectral Independence

TL;DR

This paper presents an efficient algorithm achieving near-optimal bounds for the $ ext{ell}_2$ Steinitz and prefix discrepancy problems, improving previous bounds and extending applicability under certain conditions.

Contribution

It introduces a new algorithm matching the conjectured $O( ext{d}^{1/2})$ bound for the $ ext{ell}_2$ Steinitz problem under mild conditions, using affine spectral independence and a novel data structure.

Findings

Achieves $O( ext{d}^{1/2})$ bound for $ ext{ell}_2$ Steinitz problem.

Extends results to $ ext{ell}_2$ prefix discrepancy.

Employs a new decoupling technique and a global interval tree.

Abstract

A classical result of Steinitz from 1913 \cite{Ste13}, answering an earlier question of Riemann and L\'evy (e.g., \cite{Lev05}), states that for any norm $\|\cdot\|$ in $\mathbb{R}^d$ and any set of vectors $v_1, \cdots, v_n \in \R^d$ satisfying $\sum_{i=1}^n v_i = 0$, there exists an ordering $\pi: [n] \rightarrow [n]$ such that every partial sum along this order is bounded by $O(d)$, i.e., $\big\| \sum_{i=1}^t v_{\pi(i)} \big\| \leq O(d)$ for all $t \in [n]$. Steinitz's bound is tight up to constants in general, but for the $\ell_2$ norm $\|\cdot\|_2$, it has been conjectured that the best bound is $O(\sqrt{d})$. Almost a century later, a breakthrough work of Banaszczyk \cite{Ban12} gave a bound of $O(\sqrt{d} + \sqrt{\log n})$ for the $\ell_2$ Steinitz problem, matching the conjecture under the mild assumption that $d \geq \Omega(\log n)$. Banaszczyk's result is non-constructive, and the previous best algorithmic bound was $O(\sqrt{d \log n})$, due to Bansal and Garg \cite{BG17}. In this work, we give an efficient algorithm that matches the conjectured $O(\sqrt{d})$ bound for the $\ell_2$ Steinitz problem under the slightly worse, yet still polylogarithmic, condition of $d \geq \Omega(\log^7 n)$. As in prior work, our result extends to the harder problem of $\ell_2$ prefix discrepancy. We employ the framework of obtaining the desired ordering via a discrete Brownian motion, guided by a semidefinite program (SDP). To obtain our results, we use the new technique of ``Decoupling via Affine Spectral Independence'', proposed by Bansal and Jiang \cite{BJ26} to achieve substantial progress on the Beck-Fiala and Koml\'os conjectures, together with a ``Global Interval Tree'' data structure that simultaneously controls the deviations for all prefixes.