The Complexity of Gerrymandering Over Graphs: Paths and Trees

TL;DR

This paper studies the computational complexity of gerrymandering over graph structures like paths and trees, revealing hardness results and polynomial cases, thus advancing understanding of the problem's difficulty in social network contexts.

Contribution

It classifies the NP-hardness of gerrymandering over paths and trees, providing the first complexity dichotomies for these graph classes and settling open questions.

Findings

Gerrymandering over graphs is NP-hard on paths.

The problem is polynomial-time solvable for two parties on trees.

It becomes weakly NP-hard for three parties on trees.

Abstract

Roughly speaking, gerrymandering is the systematic manipulation of the boundaries of electoral districts to make a specific (political) party win as many districts as possible. While typically studied from a geographical point of view, addressing social network structures, the investigation of gerrymandering over graphs was recently initiated by Cohen-Zemach et al. [AAMAS 2018]. Settling three open questions of Ito et al. [AAMAS 2019], we classify the computational complexity of the NP-hard problem Gerrymandering over Graphs when restricted to paths and trees. Our results, which are mostly of negative nature (that is, worst-case hardness), in particular yield two complexity dichotomies for trees. For instance, the problem is polynomial-time solvable for two parties but becomes weakly NP-hard for three. Moreover, we show that the problem remains NP-hard even when the input graph is a path.