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

TL;DR

This paper extends discrepancy theory to non-additive coverage functions, providing a polynomial bound in a sparse setting that generalizes classical results like Beck-Fiala.

Contribution

It introduces a constructive discrepancy bound for non-additive coverage functions in a sparse setting, generalizing classical discrepancy bounds.

Findings

Established a polynomial discrepancy bound in terms of t, k, and log n.

Generalized the classical Beck-Fiala theorem to non-additive coverage functions.

Provided a constructive method for discrepancy minimization in this setting.

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 $\log n$.