Discrepancy Minimization via a Self-Balancing Walk

TL;DR

This paper introduces a new simple random process for discrepancy minimization in multiple dimensions, providing tight bounds and efficient algorithms for several vector balancing problems and the Komlós conjecture.

Contribution

It analyzes a novel self-balancing walk, yielding tight discrepancy bounds and linear-time algorithms for key problems in vector balancing and the Komlós conjecture.

Findings

Tight bounds up to logarithmic factors for online vector balancing.

Linear time algorithms for the Komlós conjecture.

Analysis of a new random process for discrepancy minimization.

Abstract

We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Koml\'{o}s conjecture.