Tight Lower Bound for Multicolor Discrepancy

2025-04-25

AI Summary

This paper investigates lower bounds on the *k*-color multicolor discrepancy of hypergraphs. For *n*-vertex hypergraphs under *k* ≥ 2 colorings, it establishes the first asymptotically tight Ω(√*n*) lower bound—eliminating the log *k* factor present in prior work by Caragiannis et al. and achieving theoretical optimality. Methodologically, the proof combines probabilistic construction, refined probabilistic analysis, and hypergraph combinatorics; crucially, it employs precise discrete probability estimates to avoid the logarithmic penalties inherent in classical approaches. Notably, the bound is independent of *k*, rendering it substantially stronger than all previous results—improving upon them by a factor of approximately √log *k*. This represents the strongest known theoretical lower bound for multicolor discrepancy, with direct implications for applications such as group-fair allocation and load balancing.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

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

Problem

• Prove tight lower bound for multicolor discrepancy
• Improve previous bound from Ω(√(n/log k)) to Ω(√n)
• Apply result to enhance group fair division bounds

Innovation

• Proves tight lower bound for multicolor discrepancy
• Improves previous bound by Caragiannis et al
• Applies to group fair division bounds