Discrepancy Theory: An Algorithmic and Geometric Perspective

Abstract

Combinatorial discrepancy theory is a subject with roots in combinatorics, geometry, and number theory, and with numerous applications to mathematics and computer science. At its core, discrepancy theory is about dividing a collection of objects into two parts that are as balanced as possible. For example, given a collection of subsets of a finite universe, we may wish to color the elements with two colors so that each set is approximately evenly split. Other problems in discrepancy are more geometric in flavor, and ask for example, to assign signs to a collection of vectors, so that the sum of the signed vectors is as small as possible. Classical results, such as the Beck-Fiala theorem and Spencer's "six deviations" result, show that it is often possible to attain remarkably small discrepancy, often far smaller than what naive random colorings achieve. In recent years, discrepancy theory has undergone a transformation, driven by new algorithmic techniques and a rich interplay between probability, optimization, and convex geometry. These developments have led not only to new constructive proofs of foundational theorems, but also to several new results and research directions. This monograph aims to provide an accessible and unified introduction to these modern developments, with a focus on the core algorithmic and convex geometric ideas that have driven them. For several results, we provide new simpler analyses, while highlighting the intuition behind the proofs.