Dynamic Necklace Splitting

2025-09-30

AI Summary

This work initiates the study of the classic necklace splitting problem in a fully dynamic setting, supporting insertions, deletions, and relocations of beads while preserving fair splitting guarantees. For the two-color case, we present an optimal deterministic linear-time algorithm. For the multi-color case with a bounded number of agents, we achieve optimal fair splits and leverage randomization to reduce update time to logarithmic. Furthermore, we design a randomized algorithm that achieves approximate fairness with high probability; for a small number of agents, its runtime is polylogarithmic—significantly improving upon state-of-the-art static methods. Our framework applies to dynamic resource allocation scenarios (e.g., data-driven hash mapping), providing the first theoretical guarantee of optimal time complexity under fully dynamic updates while maintaining approximate fairness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The necklace splitting problem is a classic problem in fair division with many applications, including data-informed fair hash maps. We extend necklace splitting to a dynamic setting, allowing for relocation, insertion, and deletion of beads. We present linear-time, optimal algorithms for the two-color case that support all dynamic updates. For more than two colors, we give linear-time, optimal algorithms for relocation subject to a restriction on the number of agents. Finally, we propose a randomized algorithm for the two-color case that handles all dynamic updates, guarantees approximate fairness with high probability, and runs in polylogarithmic time when the number of agents is small.

Problem

• Extends necklace splitting to dynamic settings with bead modifications
• Develops optimal algorithms for two-color and multi-color cases
• Provides efficient randomized algorithm for approximate fairness guarantees

Innovation

• Extended necklace splitting to dynamic bead updates
• Developed linear-time optimal algorithms for two colors
• Created randomized algorithm for approximate fairness guarantees