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

Authors Tatiana Rocha Avila , Lars Rohwedder , Leo Wennmann

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LIPIcs.ESA.2026.139.pdf
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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Combinatorial Optimization
  • Discrepancy Theory

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Abstract

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.

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Tatiana Rocha Avila, Lars Rohwedder, and Leo Wennmann. Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 139:1-139:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.139

Author Details

Tatiana Rocha Avila
  • Goethe University Frankfurt, Germany
Lars Rohwedder
  • University of Southern Denmark, Odense, Denmark
Leo Wennmann
  • University of Southern Denmark, Odense, Denmark

Funding

Lars Rohwedder and Leo Wennmann: Supported by Dutch Research Council (NWO) project "The Twilight Zone of Efficiency: Optimality of Quasi-Polynomial Time Algorithms" [grant number OCEN.W.21.268].
  • Avila, Tatiana Rocha: Funded by DFG Research Unit ADYN under grant DFG 411362735.

Acknowledgements

We thank the anonymous reviewers for many helpful comments.

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