Discrepancy Beyond Additive Functions with Applications to Fair Division

TL;DR

This work generalizes discrepancy theory to non-additive, 1-Lipschitz set functions and proves a tight- up-to-log factor bound on the $k$-color discrepancy for prime $k$, independent of the ground set size. The key innovation is replacing LP-based reductions with a topological consensus division tool (Borsuk--Ulam) to obtain a fractional $k$-coloring with at most $n(k-1)$ fractional elements, followed by randomized rounding and McDiarmid concentration. The discrepancy bound directly yields fair-division guarantees: for monotone utilities, there exists a consensus halving up to $O(\sqrt{n \log n})$ goods, and hence envy-freeness up to the same bound, extending results beyond additive utilities. The results connect discrepancy and fair division through novel topological methods, highlight parallel work, and outline open questions about tightening bounds, sparsity, multi-partitions, and computational aspects.

Abstract

We consider a setting where we have a ground set $M$ together with real-valued set functions $f_1, \dots, f_n$, and the goal is to partition $M$ into two sets $S_1,S_2$ such that $|f_i(S_1) - f_i(S_2)|$ is small for every $i$. Many results in discrepancy theory can be stated in this form with the functions $f_i$ being additive. In this work, we initiate the study of the unstructured case where $f_i$ is not assumed to be additive. We show that even without the additivity assumption, the upper bound remains at most $O(\sqrt{n \log n})$. Our result has implications on the fair allocation of indivisible goods. In particular, we show that a consensus halving up to $O(\sqrt{n \log n})$ goods always exists for $n$ agents with monotone utilities. Previously, only an $O(n)$ bound was known for this setting.

Theorems & Definitions (18)

• Definition 1
• Theorem 2: Main Theorem (Informal)
• Theorem 3
• Theorem 5: SimmonsSu03FilosRatsikasHoSo21
• Theorem 6: McDiarmid89
• Definition 7
• Theorem 8: Main Theorem
• Lemma 9
• proof
• proof : Proof of \ref{['thm:main']}