An Improved Bound for the Beck-Fiala Conjecture
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913974127165440 |
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| author | Bansal, Nikhil Jiang, Haotian |
| author_facet | Bansal, Nikhil Jiang, Haotian |
| contents | In 1981, Beck and Fiala [Discrete Appl. Math, 1981] conjectured that given a set system $A \in \{0,1\}^{m \times n}$ with degree at most $k$ (i.e., each column of $A$ has at most $k$ non-zeros), its combinatorial discrepancy $\mathsf{disc}(A) := \min_{x \in \{\pm 1\}^n} \|Ax\|_\infty$ is at most $O(\sqrt{k})$. Previously, the best-known bounds for this conjecture were either $O(k)$, first established by Beck and Fiala [Discrete Appl. Math, 1981], or $O(\sqrt{k \log n})$, first proved by Banaszczyk [Random Struct. Algor., 1998].
We give an algorithmic proof of an improved bound of $O(\sqrt{k \log\log n})$ whenever $k \geq \log^5 n$, thus matching the Beck-Fiala conjecture up to $O(\sqrt{\log \log n})$ for almost the full regime of $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Improved Bound for the Beck-Fiala Conjecture Bansal, Nikhil Jiang, Haotian Combinatorics Discrete Mathematics Data Structures and Algorithms In 1981, Beck and Fiala [Discrete Appl. Math, 1981] conjectured that given a set system $A \in \{0,1\}^{m \times n}$ with degree at most $k$ (i.e., each column of $A$ has at most $k$ non-zeros), its combinatorial discrepancy $\mathsf{disc}(A) := \min_{x \in \{\pm 1\}^n} \|Ax\|_\infty$ is at most $O(\sqrt{k})$. Previously, the best-known bounds for this conjecture were either $O(k)$, first established by Beck and Fiala [Discrete Appl. Math, 1981], or $O(\sqrt{k \log n})$, first proved by Banaszczyk [Random Struct. Algor., 1998]. We give an algorithmic proof of an improved bound of $O(\sqrt{k \log\log n})$ whenever $k \geq \log^5 n$, thus matching the Beck-Fiala conjecture up to $O(\sqrt{\log \log n})$ for almost the full regime of $k$. |
| title | An Improved Bound for the Beck-Fiala Conjecture |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2508.01937 |