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

Recent concurrent work by Dupré 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 $\log n$.

💡 Research Summary

The paper investigates multicolor discrepancy for a class of non‑additive functions called coverage functions, extending the classical Beck‑Fiala setting to a non‑linear regime. A coverage function is defined by a universe U and, for each item i, a subset U_i ⊆ U; the function value on a set T of items is the number of distinct elements of U covered by the items in T. The authors focus on t‑sparse families, meaning each item appears in at most t different covering sets across all functions. This sparsity implies a 1‑Lipschitz property but destroys the linear structure that underlies traditional discrepancy tools such as Beck‑Fiala’s linear algebraic proof or partial coloring methods.

The main contribution is a polynomial‑time algorithm that, for any number of colors k, produces a coloring χ: