Parameterized Complexity of Gerrymandering

TL;DR

This paper investigates the computational complexity of gerrymandering as a graph partitioning problem, proving hardness results on trees and providing fixed-parameter tractable algorithms for special cases.

Contribution

It establishes W[2]-hardness of Gerrymandering on trees and offers an FPT algorithm when the number of leaves is fixed, advancing understanding of its parameterized complexity.

Findings

W[2]-hardness of Gerrymandering on trees with respect to number of districts

W[2]-hardness when trees have a fixed number of leaves

FPT algorithm for Gerrymandering on paths when the number of leaves is fixed

Abstract

In a representative democracy, the electoral process involves partitioning geographical space into districts which each elect a single representative. These representatives craft and vote on legislation, incentivizing political parties to win as many districts as possible (ideally a plurality). Gerrymandering is the process by which district boundaries are manipulated to the advantage of a desired candidate or party. We study the parameterized complexity of Gerrymandering, a graph problem (as opposed to Euclidean space) formalized by Cohen-Zemach et al. (AAMAS 2018) and Ito et al. (AAMAS 2019) where districts partition vertices into connected subgraphs. We prove that Unit Weight Gerrymandering is W[2]-hard on trees (even when the depth is two) with respect to the number of districts $k$. Moreover, we show that Unit Weight Gerrymandering remains W[2]-hard in trees with $\ell$ leaves with respect to the combined parameter $k+\ell$. In contrast, Gupta et al. (SAGT 2021) give an FPT algorithm for Gerrymandering on paths with respect to $k$. To complement our results and fill this gap, we provide an algorithm to solve Gerrymandering that is FPT in $k$ when $\ell$ is a fixed constant.