Redistricting Algorithms

Abstract

Why not have a computer just draw a map? This is something you hear a lot when people talk about gerrymandering, and it's easy to think at first that this could solve redistricting altogether. But there are more than a couple problems with this idea. In this chapter, two computer scientists survey what's been done in algorithmic redistricting, discuss what doesn't work and highlight approaches that show promise. This preprint was prepared as a chapter in the forthcoming edited volume Political Geometry, an interdisciplinary collection of essays on redistricting. (mggg.org/gerrybook)

Locally Fair Partitioning

“…There has been extensive work on redistricting algorithms, going back to 1960s (Hess et al 1965), for constructing contiguous, compact, and balanced districts. Many different approaches, including integer programming (Goderbauer 2014), simulated annealing (Altman and McDonald 2010), evolutionary algorithms (Liu, Cho, and Wang 2016), Voronoi diagram based methods (Svec, Burden, and Dilley 2007;Cohen-Addad, Klein, and Young 2018), MCMC methods (Bangia et al 2017;DeFord, Duchin, and Solomon 2021), have been proposed; see (Becker and Solomon 2020) for a recent survey. A line of work on redistricting algorithms focuses on combating manipulation such as gerrymandering: when district plans have been engineered to provide advantage to individual candidates or to parties (Borodin et al 2018).…”

Locally Fair Partitioning

“…There has been extensive work on redistricting algorithms, going back to 1960s [19], for constructing contiguous, compact, and balanced districts. Many different approaches, including integer programming [17], simulated annealing [1], evolutionary algorithms [22], Voronoi diagram based methods [12,29], MCMC methods [4,13], have been proposed; see [5] for a recent survey. A line of work on redistricting algorithms focuses on combating manipulation such as gerrymandering: when district plans have been engineered to provide advantage to individual candidates or to parties [6].…”

“…In many redistricting algorithms, existing methods generate maps without explicitly incorporating notions of fairness, but instead focusing on compactness. Popular methods generate an ensemble of plans and compare the number of representatives each party gets in the generated maps with the number received under the actual proposed maps [5]. In practice, political groups use many justifications for whether a plan is fair, and our paper offers a new formal model which may be used for auditing-arguing that various plans satisfy properties of fairness [13,24].…”

“…Since β < 1 − ε, π 1 must include points from a neighboring blue monochromatic interval; without loss of generality, let it intersect with B 1 . 5 Then π 1 cannot intersect R 2 ; otherwise B 1 would deviate. Therefore, π 1 includes at most βσ points from R 1 and some points from B 1 .…”

“…For the last step, the algorithm linear scans LF((0, n], i 1 , i 2 , i 3 ) for all O(ε 3 σ 3 ) possible values of (i 1 , i 2 , i 3 ), which is O(ε 3 σ 3 )-time. Therefore, the total time complexity of the algorithm is given by O(nσ 4 ) + O(ε 4 σ 6 ) + O(ε 3 σ 3 ) = O(nσ 5 ), as σ = O(n) and ε = O (1). For ε ∈ [0, 1/3), it is O(nσ 4 ) with the second term being O(ε 4 σ 4 ).…”