Constructive Algorithms for Discrepancy Minimization

TL;DR

This paper introduces polynomial-time algorithms for discrepancy minimization that match existential bounds using a novel random walk approach combined with semidefinite programming and the entropy method.

Contribution

It presents the first efficient algorithms achieving near-optimal discrepancy bounds and offers an approximation result for the discrepancy problem.

Findings

Algorithms achieve bounds similar to existential results

Random walk approach effectively minimizes discrepancy

Semidefinite programming guides the coloring process

Abstract

Given a set system (V,S), V={1,...,n} and S={S1,...,Sm}, the minimum discrepancy problem is to find a 2-coloring of V, such that each set is colored as evenly as possible. In this paper we give the first polynomial time algorithms for discrepancy minimization that achieve bounds similar to those known existentially using the so-called Entropy Method. We also give a first approximation-like result for discrepancy. The main idea in our algorithms is to produce a coloring over time by letting the color of the elements perform a random walk (with tiny increments) starting from 0 until they reach $-1$ or $+1$. At each time step the random hops for various elements are correlated using the solution to a semidefinite program, where this program is determined by the current state and the entropy method.