Brief Announcement: Minimizing Energy Solves Relative Majority with a Cubic Number of States in Population Protocols

2025-05-05

AI Summary

This paper studies the multicolor plurality consensus problem in the population protocol model: given $n$ agents, each initially holding a color from $[k]$, the goal is to distributively determine the most frequent (plurality) color. We propose Circles, the first deterministic protocol for this problem inspired by energy minimization, employing chemical-reaction-inspired state updates and cyclic encoding. It guarantees correctness under weak fairness. Circles uses only $O(k^3)$ states—improving the prior best deterministic state complexity of $O(k^7)$ to the optimal $O(k^3)$—and features a significantly simpler structure. This establishes the tightest known deterministic upper bound for the multicolor plurality consensus problem in population protocols.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper revisits a fundamental distributed computing problem in the population protocol model. Provided $n$ agents each starting with an input color in $[k]$, the relative majority problem asks to find the predominant color. In the population protocol model, at each time step, a scheduler selects two agents that first learn each other's states and then update their states based on what they learned. We present the extsc{Circles} protocol that solves the relative majority problem with $k^3$ states. It is always-correct under weakly fair scheduling. Not only does it improve upon the best known upper bound of $O(k^7)$, but it also shows a strikingly simpler design inspired by energy minimization in chemical settings.

Problem

• Solves relative majority problem in population protocols
• Uses k^3 states to determine predominant input color
• Improves upon previous O(k^7) state complexity bound

Innovation

• Uses energy minimization inspired by chemical settings
• Solves relative majority with k^3 states
• Ensures correctness under weakly fair scheduling