Tight Lower Bound for Multicolor Discrepancy

TL;DR

The paper addresses multicolor discrepancy by proving an asymptotically tight $\Omega(\sqrt{n})$ lower bound for $\mathrm{disc}^{\max}(n,k)$ that holds for all $k\ge2$, resolving a longstanding gap. The authors introduce and leverage $p$-weighted discrepancy $\mathrm{wdisc}_p$ and a simple Hadamard-based construction to achieve this lower bound, and show how it implies stronger lower bounds in group fair division (EF, PROP, CD) through new reductions. They also establish improved upper bounds for proportionality via an asymmetric discrepancy framework, including a recursive upper bound $\mathrm{asymdisc}^{\max}(n_1,\dots,n_k) \le O(\sqrt{n_1})$ and the consequence $c^{\mathrm{PROP}}(n_1,\dots,n_k) \le O(\sqrt{n_1})$, with exact improvements in balanced regimes. Overall, the work closes the main gap on multicolor discrepancy and advances the understanding of fair division with indivisible goods, while outlining open questions for tightening EF/PROP gaps and achieving tight PROP bounds in all parameter regimes.

Abstract

We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $Ω(\sqrt{n})$. This improves on the previously known lower bound of $Ω(\sqrt{n/\log k})$ due to Caragiannis et al. (arXiv:2502.10516). As an application, we show that our result implies improved lower bounds for group fair division.

Figures (1)

• Figure 1: Summary of our lower bounds for group fair division and known lower bounds. Here, we assume $n_1 \geq \cdots \geq n_k \geq 1$. As discussed below, our lower bound is a strict improvement over previous lower bounds by a factor of $\Omega(\log k)$ for all values of $n_1, \dots, n_k$. Moreover, the improvement for $c^{\text{EF}}$ and $c^{\text{PROP}}$ can be as large as $\Omega(\sqrt{k \log k})$ and $\Omega(k\sqrt{\log k})$ respectively for some values of $n_1, \dots, n_k$. Finally, note also that the previous three lower bounds for $c^{\text{PROP}}$ are incomparable.

Theorems & Definitions (24)

• Definition 1.1: DoerrSr03
• Definition 1.2
• Definition 2.1: DoerrSr03
• Theorem 2.2: LovaszSpVe86
• Lemma 3.1: Chazelle01CharikarNeNi11
• Theorem 3.2
• proof : Proof of \ref{['thm:main-weighted']}
• Corollary 3.3
• Corollary 4.1
• Corollary 4.2