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Drawing Competitive Districts in Redistricting

Gabriel Chuang, Oussama Hanguir, Clifford Stein

TL;DR

This work examines the problem of drawing redistricting plans with a fixed number of competitive districts, formalizing two competitiveness notions: δ-Vote-Band-Competitive (δ-VBC) and Swing-based competitiveness. It proves NP-hardness for maximizing the number of competitive districts even on simple grid instances, contrasting this with the practical tractability of achieving high competitiveness via a simple hill-climbing heuristic. The authors implement and test the method on real precinct graphs for North Carolina and Arizona, showing plans where all districts can be competitive under various thresholds and discussing trade-offs with compactness and population balance. The results demonstrate that, despite theoretical hardness, near-optimal competitive districtings are feasible in real-world settings, informing policy discussions on competitiveness, responsiveness, and district design.

Abstract

In the process of redistricting, one important metric is the number of competitive districts, that is, districts where both parties have a reasonable chance of winning a majority of votes. Competitive districts are important for achieving proportionality, responsiveness, and other desirable qualities; some states even directly list competitiveness in their legally-codified districting requirements. In this work, we discuss the problem of drawing plans with at least a fixed number of competitive districts. In addition to the standard, ``vote-band'' measure of competitivenesss (i.e., how close was the last election?), we propose a measure that explicitly considers ``swing voters'' - the segment of the population that may choose to vote either way, or not vote at all, in a given election. We present two main, contrasting results. First, from a computational complexity perspective, we show that the task of drawing plans with competitive districts is NP-hard, even on very natural instances where the districting task itself is easy (e.g., small rectangular grids of population-balanced cells). Second, however, we show that a simple hill-climbing procedure can in practice find districtings on real states in which all the districts are competitive. We present the results of the latter on the precinct-level graphs of the U.S. states of North Carolina and Arizona, and discuss trade-offs between competitiveness and other desirable qualities.

Drawing Competitive Districts in Redistricting

TL;DR

This work examines the problem of drawing redistricting plans with a fixed number of competitive districts, formalizing two competitiveness notions: δ-Vote-Band-Competitive (δ-VBC) and Swing-based competitiveness. It proves NP-hardness for maximizing the number of competitive districts even on simple grid instances, contrasting this with the practical tractability of achieving high competitiveness via a simple hill-climbing heuristic. The authors implement and test the method on real precinct graphs for North Carolina and Arizona, showing plans where all districts can be competitive under various thresholds and discussing trade-offs with compactness and population balance. The results demonstrate that, despite theoretical hardness, near-optimal competitive districtings are feasible in real-world settings, informing policy discussions on competitiveness, responsiveness, and district design.

Abstract

In the process of redistricting, one important metric is the number of competitive districts, that is, districts where both parties have a reasonable chance of winning a majority of votes. Competitive districts are important for achieving proportionality, responsiveness, and other desirable qualities; some states even directly list competitiveness in their legally-codified districting requirements. In this work, we discuss the problem of drawing plans with at least a fixed number of competitive districts. In addition to the standard, ``vote-band'' measure of competitivenesss (i.e., how close was the last election?), we propose a measure that explicitly considers ``swing voters'' - the segment of the population that may choose to vote either way, or not vote at all, in a given election. We present two main, contrasting results. First, from a computational complexity perspective, we show that the task of drawing plans with competitive districts is NP-hard, even on very natural instances where the districting task itself is easy (e.g., small rectangular grids of population-balanced cells). Second, however, we show that a simple hill-climbing procedure can in practice find districtings on real states in which all the districts are competitive. We present the results of the latter on the precinct-level graphs of the U.S. states of North Carolina and Arizona, and discuss trade-offs between competitiveness and other desirable qualities.
Paper Structure (32 sections, 9 theorems, 25 equations, 7 figures)

This paper contains 32 sections, 9 theorems, 25 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Competitiveness of districting plans.
  5. Computational complexity of redistricting.
  6. Preliminaries and Problem Formulation
  7. Voting setting
  8. Districtings and Competitive Districts
  9. Competitiveness
  10. Maximizing Competitive Districts
  11. Hardness results
  12. Discussion of Complexity Results
  13. Algorithms, Heuristics, and Experiment Setup
  14. Data
  15. Voter types
  16. ...and 17 more sections

Key Result

Theorem 1

For all positive $\varepsilon < \frac{1}{6}, \delta < \frac{1}{2}$, the $\delta$-VBC $\varepsilon$-valid District Maximization Problem ($\delta$-VBC-Max($\varepsilon$)) is NP-hard, even on instaces where $G$ is a grid, $d=2$, and $\varepsilon$-valid districtings (that are not $\delta$-VBC-maximal) a

Figures (7)

  • Figure 1: The chain of reductions from Subset Sum to FPSP to $\delta$-VBC-Max, for $\delta = 0.1, \varepsilon=\frac{1}{6}$. Given the Subset Sum instance $T$ (left), we construct an instance of FPSP (center) where bin $i$ has $6+t_i$ type-A units and $4-t_i$ type-B units. This induces a districting instance (right), where the central row has Party A and Party B voters corresponding to type-A and type-B units. The green and orange districts (where each district has population in $(\frac{1}{2} \pm \varepsilon)pop_{\text{total}}$ and exactly 60% Party A voters) correspond to a FPSP partition (where each partition has exactly 60% type-A units), which in turn corresponds to a solution to the original Subset Sum instance.
  • Figure 2: Precinct-level voting data for Arizona and North Carolina. Nodes are colored according to the margin in the 2020 presidential election.
  • Figure 3: Explicitly considering competitiveness allows us to draw plans with significantly fewer safe districts than both the enacted plan and a sample of compact plans.
  • Figure 4: Our simple hill-climbing procedure successfully finds plans for North Carolina where all fourteen districts are competitive.
  • Figure 5: The EI table for votes cast in Alamance County, North Carolina, Precinct 01. The inner cell values are estimated using the R package nslphom, using the known marginal values. In this case, the estimated number of swing voters is $3.55 + 9.24 + \frac{1}{2}(71.30 + 73.76 + 0.0 + 272.07) = 221.36$
  • ...and 2 more figures

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3: $\delta$-VBC-Max
  • Definition 4: Swing-Max
  • Theorem 1: Vote-Band Hardness
  • Theorem 2: Swing Hardness
  • Definition 5: FPSP($\delta$)
  • Theorem 3: FPSP Hardness
  • proof
  • Corollary 3: Hardness for more than two districts
  • ...and 14 more