Fair redistricting is hard

TL;DR

Deciding the existence of a fair redistricting that meets criteria like compactness and representation is computationally NP-hard, highlighting the complexity of creating equitable electoral maps.

Contribution

This paper proves that determining a fair redistricting map under simplified legal and fairness criteria is NP-hard, advancing theoretical understanding of gerrymandering challenges.

Findings

Deciding fair redistricting is NP-hard.

Simplified fairness criteria include geographic compactness and voter representation.

The proof is inspired by NP-hardness results in planar k-means clustering.

Abstract

Gerrymandering is a long-standing issue within the U.S. political system, and it has received scrutiny recently by the U.S. Supreme Court. In this note, we prove that deciding whether there exists a fair redistricting among legal maps is NP-hard. To make this precise, we use simplified notions of "legal" and "fair" that account for desirable traits such as geographic compactness of districts and sufficient representation of voters. The proof of our result is inspired by the work of Mahanjan, Minbhorkar and Varadarajan that proves that planar k-means is NP-hard.