Revisit the Partial Coloring Method: Prefix Spencer and Sampling

TL;DR

This paper advances discrepancy theory by dissecting the partial coloring method through two complementary algorithmic lenses: a linear-algebraic framework and a Gaussian measure framework. It proves that, for $A\in\{0,1\}^{m\times n}$ with $n\ge m$, one can obtain a prefix coloring with prefix discrepancy $O(\sqrt{m})$ while fixing $\Omega(n)$ entries, and recursively achieve a full coloring with prefix discrepancy $O(\sqrt{m}\cdot\log\frac{O(n)}{m})$; it also shows a Gaussian-measure-based approach yielding the same prefix bound with $\Omega(n)$ fixed entries and a full coloring within the same order up to a logarithmic factor, underpinned by small-deviation bounds for Gaussian processes. Beyond the Spencer setting, the authors extend the linear-algebraic framework to a sampling procedure with min-entropy $\Omega(n)$ via leverage scores, enabling diverse colorings and applying to Beck-Fiala and vector balancing. They further adapt these ideas to prefix discrepancy, online discrepancy, and matrix discrepancy contexts, highlighting practical paths to robust, randomized colorings with strong discrepancy guarantees. The work thus unifies and extends algorithmic methods in discrepancy theory, providing near-optimal prefix-discrepancy results and practical sampling techniques with provable entropy properties. This yields more versatile tools for discrepancy minimization and applications like Beck-Fiala, prefix discrepancy, and matrix sparsification, with implications for randomized design and online settings.

Abstract

As the most powerful tool in discrepancy theory, the partial coloring method has wide applications in many problems including the Beck-Fiala problem and Spencer's celebrated result. Currently, there are two major algorithmic methods for the partial coloring method: the first approach uses linear algebraic tools; and the second is called Gaussian measure algorithm. We explore the advantages of these two methods and show the following results for them separately. 1. Spencer conjectured that the prefix discrepancy of any $\mathbf{A} \in \{0,1\}^{m \times n}$ is $O(\sqrt{m})$. We show how to find a partial coloring with prefix discrepancy $O(\sqrt{m})$ and $Ω(n)$ entries in $\{ \pm 1\}$ efficiently. To the best of our knowledge, this provides the first partial coloring whose prefix discrepancy is almost optimal. However, unlike the classical discrepancy problem, there is no reduction on the number of variables $n$ for the prefix problem. By recursively applying partial coloring, we obtain a full coloring with prefix discrepancy $O(\sqrt{m} \cdot \log \frac{O(n)}{m})$. Prior to this work, the best bounds of the prefix Spencer conjecture for arbitrarily large $n$ were $2m$ and $O(\sqrt{m \log n})$. 2. Our second result extends the first linear algebraic approach to a sampling algorithm in Spencer's classical setting. On the first hand, Spencer proved that there are $1.99^m$ good colorings with discrepancy $O(\sqrt{m})$. Hence a natural question is to design efficient random sampling algorithms in Spencer's setting. On the other hand, some applications of discrepancy theory, prefer a random solution instead of a fixed one. Our second result is an efficient sampling algorithm whose random output has min-entropy $Ω(n)$ and discrepancy $O(\sqrt{m})$. Moreover, our technique extends the linear algebraic framework by incorporating leverage scores of randomized matrix algorithms.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 3.1: Lévy's inequality
• Claim 3.2
• Theorem 4.1
• Corollary 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5