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

T. R. Avila, Lars Rohwedder, Leo Wennmann

TL;DR

The paper studies non-additive discrepancy for coverage functions in a Beck-Fiala-type sparse setting. It introduces t-sparse coverage-function families and grounds the approach in the multilinear extension, concentration bounds, and rounding-in-expectation to derive a constructive k-coloring. For the small-sets regime it achieves a discrepancy of O(√(t^3 k) · ln(ntk)), while for big sets it attains zero discrepancy via a constructive Lovász Local Lemma argument; the general result combines these regimes to handle all set sizes. This advances constructive bounds in non-additive discrepancy and has potential applications in fair division and related combinatorial optimization tasks.

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 $\log n$.

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

TL;DR

The paper studies non-additive discrepancy for coverage functions in a Beck-Fiala-type sparse setting. It introduces t-sparse coverage-function families and grounds the approach in the multilinear extension, concentration bounds, and rounding-in-expectation to derive a constructive k-coloring. For the small-sets regime it achieves a discrepancy of O(√(t^3 k) · ln(ntk)), while for big sets it attains zero discrepancy via a constructive Lovász Local Lemma argument; the general result combines these regimes to handle all set sizes. This advances constructive bounds in non-additive discrepancy and has potential applications in fair division and related combinatorial optimization tasks.

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 items covers only elements across all functions, we prove a constructive discrepancy bound that is polynomial in , the number of colors , and .
Paper Structure (14 sections, 24 theorems, 17 equations)

This paper contains 14 sections, 24 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Sparse Coverage Functions
  3. Our Contribution
  4. Future Directions
  5. Preliminaries
  6. Multilinear Extension
  7. Concentration Bounds
  8. Rounding in Expectation
  9. Sparse Coverage Functions with Small Sets
  10. Computing a k-Coloring with Few Fractional Elements
  11. Rounding a k-Coloring with Few Fractional Elements
  12. Sparse Coverage Functions with Big Sets
  13. Sparse Coverage Functions with Any Size Sets
  14. Appendix

Key Result

Theorem 2

For any prime power $k = p^{\nu}$ and any family of 1-Lipschitz functions $\mathcal{F} = \lbrace f_1, \dotsc , f_m: \lbrace0,1\rbrace^n \rightarrow \mathbb R\rbrace$, there exists a $k$-coloring of discrepancy $\mathcal{O}(\sqrt{m \log (mk)})$.

Theorems & Definitions (30)

  • Definition 1
  • Theorem 2: DupreTF25
  • Theorem 3
  • Definition 3: $t$-Sparse Family of Coverage Functions
  • Theorem 4
  • Theorem 5
  • Definition 6: Multilinear Extension
  • Definition 7: Restricted Multilinear Extension
  • Lemma 7
  • Lemma 8: McDiarmid's Inequality mcdiarmid1989method
  • ...and 20 more