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

TL;DR

This paper develops mixed integer linear programming models to optimize nonconvex objectives in computational redistricting, providing tight bounds and effective solutions for estimating political representation.

Contribution

It introduces novel linear approximations for nonconvex redistricting objectives, extending previous work on contiguity constraints to improve optimization accuracy.

Findings

Models produce tight bounds on redistricting problems.

Approaches are effective on county-level data.

Enhanced optimization methods for political representation estimation.

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.