The Balanced Up-Down Walk

2026-02-12

AI Summary

This work addresses the challenge of efficiently generating random graph partitions that satisfy exact population balance constraints in political redistricting. The authors propose a novel Markov chain, termed the Balanced Up-Down (BUD) Walk, which performs random walks over spanning trees while preserving the property that the tree can be split into balanced subtrees. This approach achieves irreducible sampling over the space of exactly balanced partitions—a property not guaranteed by existing methods such as ReCom—and ensures connectivity of the state space even when ReCom is reducible. Theoretical analysis establishes the irreducibility of the BUD Walk under various conditions, and empirical results demonstrate its superior sampling quality and exploration capability. Additionally, the study improves the algorithm for determining balanced subtree splittability and proves that the associated counting problem is #P-complete.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Markov chains based on spanning trees have been hugely influential in algorithms for assessing fairness in political redistricting. The input graph represents the geographic building blocks of a jurisdiction. The goal is to output a large ensemble of random graph partitions, which is done by drawing and splitting random spanning trees. Crucially, these subtrees must be balanced, since political districts are required to have equal population. The Up-Down walk (on trees or forests) repeatedly adds a random edge then deletes a random edge to produce a new tree or forest; it can be used to efficiently generate a large ensemble, but the rejection rate to maintain balance grows exponentially with the number of parts. ReCom, the most widely-used class of Markov chains, circumvents this complexity barrier by merging and splitting pairs of districts at a time. This runs fast in practice but can have trouble exploring the state space. To overcome these efficiency and mixing barriers, we propose a new Markov chain called the Balanced Up-Down (BUD) walk. The main idea is to run the Up-Down walk on the space of trees, but require all steps to preserve the property that the tree is splittable into balanced subtrees. The BUD walk samples from a known invariant measure under exact balance. We prove that the BUD walk is irreducible in several cases, including a regime where ReCom is not irreducible. Running the BUD walk efficiently presents algorithmic challenges, especially when parts are allowed to deviate from their ideal size. A key subroutine is determining whether a tree is splittable into approximately-balanced subtrees. We give an improved analysis of an existing algorithm for this problem and prove that the associated counting problem is #P-complete. We empirically validate the usefulness of the BUD walk by comparing its performance to that of other existing methods for sampling partitions.

Problem

• balanced graph partitioning
• Markov chain
• political redistricting
• spanning trees
• population balance

Innovation

• Balanced Up-Down walk
• Markov chain
• graph partitioning
• political redistricting
• balanced spanning trees