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Balanced Spanning Tree Distributions Have Separation Fairness

Harry Chen, Kamesh Munagala, Govind S. Sankar

TL;DR

This work studies separation fairness for balanced spanning-tree redistricting distributions. It introduces the λ-smooth variant, where partitions P are sampled with probability proportional to $ ext{sp}(P)\,e^{-\lambda\cdot\text{imb}(P)}$, and proves a constant-fairness result for 2-partitions on $m\times n$ grids, with imbalance shrinking as the grid grows. The key technical advance is the unseparating-mapping framework, built on a modified Wilson algorithm and a novel walk-collapse construction, which connects local cycle changes in the dual graph to balanced partitions while preserving fairness at a granular level. The results imply that MCMC methods like ReCom maintain separation fairness in practice and offer a theoretical basis for their empirical robustness on real-world maps. The work also develops graph-theoretic tools for loop-erased random walks and dual-cycle analyses that could interest broader studies of random partitions and planar graphs.

Abstract

Sampling-based methods such as ReCom are widely used to audit redistricting plans for fairness, with the balanced spanning tree distribution playing a central role since it favors compact, contiguous, and population-balanced districts. However, whether such samples are truly representative or exhibit hidden biases remains an open question. In this work, we introduce the notion of separation fairness, which asks whether adjacent geographic units are separated with at most a constant probability (bounded away from one) in sampled redistricting plans. Focusing on grid graphs and two-district partitions, we prove that a smooth variant of the balanced spanning tree distribution satisfies separation fairness. Our results also provide theoretical support for popular MCMC methods like ReCom, suggesting that they maintain fairness at a granular level in the sampling process. Along the way, we develop tools for analyzing loop-erased random walks and partitions that may be of independent interest.

Balanced Spanning Tree Distributions Have Separation Fairness

TL;DR

This work studies separation fairness for balanced spanning-tree redistricting distributions. It introduces the λ-smooth variant, where partitions P are sampled with probability proportional to , and proves a constant-fairness result for 2-partitions on grids, with imbalance shrinking as the grid grows. The key technical advance is the unseparating-mapping framework, built on a modified Wilson algorithm and a novel walk-collapse construction, which connects local cycle changes in the dual graph to balanced partitions while preserving fairness at a granular level. The results imply that MCMC methods like ReCom maintain separation fairness in practice and offer a theoretical basis for their empirical robustness on real-world maps. The work also develops graph-theoretic tools for loop-erased random walks and dual-cycle analyses that could interest broader studies of random partitions and planar graphs.

Abstract

Sampling-based methods such as ReCom are widely used to audit redistricting plans for fairness, with the balanced spanning tree distribution playing a central role since it favors compact, contiguous, and population-balanced districts. However, whether such samples are truly representative or exhibit hidden biases remains an open question. In this work, we introduce the notion of separation fairness, which asks whether adjacent geographic units are separated with at most a constant probability (bounded away from one) in sampled redistricting plans. Focusing on grid graphs and two-district partitions, we prove that a smooth variant of the balanced spanning tree distribution satisfies separation fairness. Our results also provide theoretical support for popular MCMC methods like ReCom, suggesting that they maintain fairness at a granular level in the sampling process. Along the way, we develop tools for analyzing loop-erased random walks and partitions that may be of independent interest.

Paper Structure

This paper contains 58 sections, 17 theorems, 24 equations, 75 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Redistricting Plans and the Spanning Tree Distribution
  3. The ReCom Markov Chain.
  4. Separation Fairness and Our Result
  5. Main Result.
  6. Connection to ReCom.
  7. Technical Contribution.
  8. Related Work
  9. Redistricting as Optimization.
  10. Redistricting as Ensembles.
  11. Separation Fairness.
  12. Scaling Limits.
  13. Preliminaries
  14. Key Definitions for Redistricting
  15. The Spanning Tree Distribution via Random Walks
  16. ...and 43 more sections

Key Result

theorem 1

For any constant $\lambda \geq 0$, there is a constant $f_{\lambda} > 0$ such that for the distribution $\lambda$-$\mathcal{D}_{\mathtt{sp}}$ on a $n \times m$ grid (with $m,n = \Omega(1)$) with $k = 2$ is $f_{\lambda}$-fair.

Figures (75)

  • Figure 1: Histogram of separation probabilities of the endpoints of edges in three different graphs over 1000 redistricting plans sampled from ReCom with a population imbalance of $1\%$.
  • Figure 2: A walk from $v_1$ to $v_4$. The numbers above an edge represent the indices of the sequence where that edge appears. That is, the walk is $W=v_1(v_1,v_2)v_2(v_2,v_3)v_3(v_3,v_4)v_4(v_4,v_2)v_2(v_2,v_3)v_3(v_3,v_4)v_4$. The only loop of $W$ is drawn dashed in red. That is, the walk $v_2(v_2,v_3)v_3(v_3,v_4)v_4(v_4,v_2)v_2$ is a loop of $W$ whereas $v_3(v_3,v_4)v_4(v_4,v_2)v_2(v_2,v_3)v_3$ is not. The loop erasure is the directed path $\{(v_1,v_2),(v_2,v_3),(v_3,v_4)\}$.
  • Figure 3: An example of a random walk $\pi$ (shown in solid black) started from $x^*$ to $y^*$. The shaded region is the interior of the cycle $\pi\cup \{\langle x^*,y^*\rangle\}$. Here, $\langle x^*,y^*\rangle=\mathrm{dual}((u,v))$. The grey squares are dual nodes, and the black circles are primal nodes.
  • Figure 4: Paths that are locally different on a subgrid $H^*$ (drawn with a dotted boundary). The red and blue portions are part of exactly one path, whereas the black portions are part of both. Note that if $H^*$ was not adjacent (and part of) the boundary of the grid, then the paths cannot differ on any edges to $v_0^*$.
  • Figure 5: Two separating cycles $C_1,C_2$ that could potentially map to the same unseparating cycle $C_3$. In this case, $\delta=1$ since $\abs{ \texttt{imb}(C_2)- \texttt{imb}(C_3)}=1$. We also have $\beta=\max(12,2)$ since edges are modified in a $3\times 3$ subgrid to go from $C_1$ to $C_3$ and two cycles map to $C_3$.
  • ...and 70 more figures

Theorems & Definitions (66)

  • Definition 1
  • theorem 1
  • theorem 2: Informal version of \ref{['thm:main-reconnect']}
  • corollary 3: Informal version of \ref{['thm:group-separation']}
  • Definition 2: Dual Graph
  • Definition 3: Imbalance $\texttt{imb}(\cdot)$
  • Definition 4: Spanning Tree Score and $\texttt{sp}(\cdot)$
  • Definition 5: Walk diestel
  • Definition 6: Loop-Erasure
  • Definition 7: Dual Cycle of a 2-Partition
  • ...and 56 more