A Uniformly Random Solution to Algorithmic Redistricting

TL;DR

The paper addresses the problem of evaluating gerrymandering by constructing a uniformly random sampler for the space of all possible $k$-partitions of a bounded-degree planar graph, enabling truly unbiased ensembles of redistricting plans. The approach builds on transfer-matrix enumeration and self-avoiding walks to count and sample partitions split by frontier states, achieving a subexponential running time of $2^{O(\sqrt{n}\log n)}$ for planar graphs with bounded degree. It provides extensions to enforce compactness and (in principle) population balance, though population constraints increase complexity and may rely on rejection sampling or binning. Experimental results on Wisconsin and Nebraska maps illustrate the enormous scale of valid partitions and demonstrate the method as a principled benchmark for uniform redistricting sampling, with potential to complement or calibrate MCMC-based approaches. The work offers a rigorous, uniform baseline for evaluating redistricting samplers and enables more robust outlier analyses in gauging gerrymandering effects.

Abstract

The process of drawing electoral district boundaries is known as political redistricting. Within this context, gerrymandering is the practice of drawing these boundaries such that they unfairly favor a particular political party, often leading to unequal representation and skewed electoral outcomes. One of the few ways to detect gerrymandering is by algorithmically sampling redistricting plans. Previous methods mainly focus on sampling from some neighborhood of ``realistic' districting plans, rather than a uniform sample of the entire space. We present a deterministic subexponential time algorithm to uniformly sample from the space of all possible $ k $-partitions of a bounded degree planar graph, and with this construct a sample of the entire space of redistricting plans. We also give a way to restrict this sample space to plans that match certain compactness and population constraints at the cost of added complexity. The algorithm runs in $ 2^{O(\sqrt{n}\log n)} $ time, although we only give a heuristic implementation. Our method generalizes an algorithm to count self-avoiding walks on a square to count paths that split general planar graphs into $ k $ regions, and uses this to sample from the space of all $ k $-partitions of a planar graph.

Figures (7)

• Figure 1: The naïve call graph for well formed bracket expressions.
• Figure 2: The DP and TM call graph for well formed bracket expressions.
• Figure 3: A partial self avoiding walk on a grid, with possible update steps on the right. Figure from grid_walks.
• Figure 4: An example of splitting a map (represented by a disc) into 5 parts via 2 SAWs (in blue) with 2 splits. Also shown is a potential frontier which would have two edges labeled $1$ as a red dashed line. These are the edges where the frontier intersects the SAW.
• Figure 5: Randomly generated bipartitions of the census tracts of the union of Wisconsin congressional districts 4 and 5 without compactness constraints.