Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting

2026-02-10

AI Summary

This work addresses the challenge of discrepancy minimization in non-additive settings—such as covering functions—where classical approaches like the Beck–Fiala theorem fail to yield tight bounds. For the first time, the authors extend the sparse Beck–Fiala framework to non-additive covering functions, encompassing canonical cases including partition matroid rank functions and edge coverings in graphs. By employing combinatorial constructions and probabilistic analysis—bypassing traditional linear algebraic tools—they constructively achieve a discrepancy bound that is a polynomial function of the sparsity parameter \( t \), the number of colors \( k \), and \( \log n \), for any \( n \)-element covering system in which each set covers at most \( t \) elements. This result overcomes a fundamental theoretical barrier in controlling discrepancy for non-additive settings.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Security and Privacy: Applications of cryptography

Abstract

Recent concurrent work by Dupr\'{e} la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math'81], it remains widely open whether comparable non-additive bounds can be achieved. Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the $n$ items covers only $t$ elements across all functions, we prove a constructive discrepancy bound that is polynomial in $t$, the number of colors $k$, and $\log n$.

Problem

• non-additive discrepancy
• coverage functions
• Beck-Fiala
• fair division
• discrepancy theory

Innovation

• non-additive discrepancy
• coverage functions
• Beck-Fiala
• constructive bound
• sparse setting