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Optimizing Representation in Redistricting: Dual Bounds for Partitioning Problems with Non-Convex Objectives

Jamie Fravel, Robert Hildebrand, Nicholas Goedert, Laurel Travis, Matthew Pierson

TL;DR

By developing mixed integer linear programming models that closely approximate the parent nonlinear model, these approaches yield tight bounds on these optimization problems and exhibit the effectiveness of these approaches on county-level data.

Abstract

We investigate optimization models for the purpose of computational redistricting. Our focus is on nonconvex objectives for estimating expected Black Representatives and Political Representation. The objectives are a composition of a ratio of variables and a normal distribution's cumulative distribution function (or ``probit curve"). We extend the work of Validi et al.~\cite{validi2022imposing}, which presented a robust implementation of contiguity constraints. By developing mixed integer linear programming models that closely approximate the parent nonlinear model, our approaches yield tight bounds on these optimization problems. We exhibit the effectiveness of these approaches on county-level data.

Optimizing Representation in Redistricting: Dual Bounds for Partitioning Problems with Non-Convex Objectives

TL;DR

By developing mixed integer linear programming models that closely approximate the parent nonlinear model, these approaches yield tight bounds on these optimization problems and exhibit the effectiveness of these approaches on county-level data.

Abstract

We investigate optimization models for the purpose of computational redistricting. Our focus is on nonconvex objectives for estimating expected Black Representatives and Political Representation. The objectives are a composition of a ratio of variables and a normal distribution's cumulative distribution function (or ``probit curve"). We extend the work of Validi et al.~\cite{validi2022imposing}, which presented a robust implementation of contiguity constraints. By developing mixed integer linear programming models that closely approximate the parent nonlinear model, our approaches yield tight bounds on these optimization problems. We exhibit the effectiveness of these approaches on county-level data.
Paper Structure (46 sections, 27 theorems, 131 equations, 19 figures, 14 tables, 2 algorithms)

This paper contains 46 sections, 27 theorems, 131 equations, 19 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background on Redistricting
  3. Integer Programming used for Algorithmic Redistricting
  4. Contributions
  5. On Legality and Compactness of Maps.
  6. Outline
  7. Redistricting Optimization Models
  8. Contiguity
  9. The Expected Black Representation Objective
  10. Including Hispanic Population
  11. The Expected Partisan Representation Objective
  12. MILP Relaxations and Approximations via Graph or Hypograph
  13. Mixed Integer Approximations for Single Ratio Objectives
  14. Standard Piecewise-Linear Approximation
  15. Linear approximation of PWL: VA-PWL model
  16. ...and 31 more sections

Key Result

Proposition 1

Suppose ${\phi}_j: \mathscr{D} \to \mathbb R$ for each $j \in M$ and consider a set $\mathscr{A}_j\subset\mathscr{D}\!\times\!\mathbb R^{{m}}$. If either $\mathop{\mathrm{gra}}\nolimits({\phi}_j) \subseteq \mathscr{A}_j$ or $\mathop{\mathrm{hyp}}\nolimits({\phi}_j) \subseteq \mathscr{A}_j$, then

Figures (19)

  • Figure 1: Left: $\mathscr{A}$ is an $\varepsilon$-approximation of $\mathop{\mathrm{gra}}\nolimits({\phi})$. Right: $\mathscr{R}$ is an $\varepsilon$-relaxation of $\mathop{\mathrm{gra}}\nolimits({\phi})$
  • Figure 2: A visual comparison of breakpoint selection methods
  • Figure 3: An example logarithmic embedding. Four segments are encoded with only two binary variables.
  • Figure 4: Computed in Mathematica, this plot shows the result of minimizing or maximizing ${{\color{blue} f }}$ over $\mathop{\mathrm{logE}}\nolimits(\psi)$ as projected onto $(r=\frac{{{\color{blue} y }}}{{{\color{blue} z }}},{{\color{blue} f }})$. For a particular $({{\color{blue} y }},{{\color{blue} z }})$, approximation \ref{['tech:LogE']} may give any value of $f$ between the red and pink lines over the ratio $\frac{{{\color{blue} y }}}{{{\color{blue} z }}}$. Importantly, this example has $\check{s} = 100$, $\hat{s}=1000$, and $\{b_k\}$ evenly distributed between $0$ and $0.55$. Each of the points evaluated has an identical distance from the origin: ${{\color{blue} y }}^2+{{\color{blue} z }}^2 = 300$.
  • Figure 5: A demonstration of Algorithm \ref{['alg:BNFull']}. The initial point $(\overline{{\color{blue} z }},\overline{{\color{blue} y }})$ goes through $3$ rotations with a mirroring at $k=2$ because $\tilde{{\color{blue} \eta }}_2 < 0$. This gives $\boldsymbol \delta = (0,1,0)$ and $q = 3$. Therefore, by Proposition \ref{['prop:BN-graycode']}, we know that $\frac{\overline{{\color{blue} y }}}{\overline{{\color{blue} z }}}$ lies within $\left(\tan\!\left(\tfrac{\pi}{8}\right), \tan\!\left(\tfrac{\pi}{8}+\frac{\pi}{32}\right)\right]$.
  • ...and 14 more figures

Theorems & Definitions (60)

  • Definition 1: Graph and Hypograph
  • Definition 2: Graph and Hypograph Reformulations
  • Proposition 1
  • proof
  • Definition 3: $\varepsilon$-Relaxation and $\varepsilon$-Approximation
  • Proposition 2: Model Error
  • proof
  • Definition 4: Piecewise Linear Approximations
  • Proposition 3
  • Proposition 4
  • ...and 50 more