Fast Discrepancy Minimization with Hereditary Guarantees

TL;DR

This paper introduces a faster algorithm for low discrepancy colorings in set systems with hereditary guarantees, improving computational efficiency and practical performance over previous methods.

Contribution

The authors present a significantly faster algorithm with hereditary guarantees for discrepancy minimization, based on new structural insights, and demonstrate its effectiveness through implementation and experiments.

Findings

Algorithm runs in $O(mn^2\,\lg(2 + m/n) + n^3)$ time.

Computes better colorings than random in practical experiments.

Handles set systems with thousands of sets efficiently.

Abstract

Efficiently computing low discrepancy colorings of various set systems, has been studied extensively since the breakthrough work by Bansal (FOCS 2010), who gave the first polynomial time algorithms for several important settings, including for general set systems, sparse set systems and for set systems with bounded hereditary discrepancy. The hereditary discrepancy of a set system, is the maximum discrepancy over all set systems obtainable by deleting a subset of the ground elements. While being polynomial time, Bansal's algorithms were not practical, with e.g. his algorithm for the hereditary setup running in time $\Omega(m n^{4.5})$ for set systems with $m$ sets over a ground set of $n$ elements. More efficient algorithms have since then been developed for general and sparse set systems, however, for the hereditary case, Bansal's algorithm remains state-of-the-art. In this work, we give a significantly faster algorithm with hereditary guarantees, running in $O(mn^2\lg(2 + m/n) + n^3)$ time. Our algorithm is based on new structural insights into set systems with bounded hereditary discrepancy. We also implement our algorithm and show experimentally that it computes colorings that are significantly better than random and finishes in a reasonable amount of time, even on set systems with thousands of sets over a ground set of thousands of elements.