Parameterized Complexity of Gerrymandering

Andrew Fraser, Brian Lavallee, Blair D. Sullivan

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 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 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 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.