A Cycle Walk for Sampling Measures on Spanning Forests for Redistricting

2025-09-10

AI Summary

Efficiently sampling connected, balanced graph partitions—critical for fair political redistricting—is hindered by high rejection rates and slow mixing in conventional MCMC methods due to constraints on connectivity and district-size balance. Method: This paper introduces Cycle Walk, a novel MCMC algorithm that operates directly on the space of valid partitions by leveraging spanning-forest structures and constructing reversible cyclic tours within the constrained state space—thereby avoiding invalid proposals and accelerating convergence. Contribution/Results: Cycle Walk achieves significantly faster mixing and higher sampling efficiency under target distributions (e.g., compactness-weighted uniform distributions) compared to state-of-the-art baselines. It demonstrates rapid convergence and high sample quality across diverse real-world and synthetic maps. Its core innovation lies in embedding structural constraints—connectivity and balance—directly into the transition mechanism, enabling precise and scalable exploration of high-dimensional constrained partition spaces. This provides a computationally tractable foundation for rigorous, fairness-aware redistricting analysis.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We introduce a new Markov Chain called the Cycle Walk for sampling measures of graph partitions where the partition elements have roughly equal size. Such Markov Chains are of current interest in the generation and evaluation of political districts. We present numerical evidence that this chain can efficiently sample target distributions that have been difficult for existing sampling Markov chains.

Problem

• Sampling balanced graph partitions efficiently
• Generating and evaluating political redistricting plans
• Overcoming limitations of existing Markov chain methods

Innovation

• Cycle Walk Markov Chain for sampling
• Graph partitions with equal size elements
• Efficient sampling of difficult target distributions