A Uniformly Random Solution to Algorithmic Redistricting

The process of drawing electoral district boundaries is known as politicalredistricting. Within this context, gerrymandering is the practice of drawingthese boundaries such that they unfairly favor a particular political party,often leading to unequal representation and skewed electoral outcomes. One ofthe few ways to detect gerrymandering is by algorithmically samplingredistricting plans. Previous methods mainly focus on sampling from someneighborhood of “realistic' districting plans, rather than a uniform sample ofthe entire space. We present a deterministic subexponential time algorithm touniformly sample from the space of all possible k-partitions of a boundeddegree planar graph, and with this construct a sample of the entire space ofredistricting plans. We also give a way to restrict this sample space to plansthat match certain compactness and population constraints at the cost of addedcomplexity. The algorithm runs in 2^O(√(n)log n) time, although weonly give a heuristic implementation. Our method generalizes an algorithm tocount self-avoiding walks on a square to count paths that split general planargraphs into k regions, and uses this to sample from the space of all k-partitions of a planar graph.