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.

翻译：在代议制民主中，选举过程涉及将地理空间划分为若干选区，每个选区选举一名代表。这些代表起草并表决立法，促使政党尽可能赢得更多选区（理想情况下获得多数席位）。选区划分不公是指操纵选区边界以有利于特定候选人或政党的过程。我们研究了选区划分不公的参量化复杂性，该问题由Cohen-Zemach等人（AAMAS 2018）和Ito等人（AAMAS 2019）形式化为图论问题（而非欧几里得空间），其中选区将顶点划分为连通子图。我们证明：单位权重选区划分不公问题在树形图中（即使深度为2时）关于选区数量$k$是W[2]-难的。此外，我们表明单位权重选区划分不公问题在具有$\ell$个叶子的树形图中关于组合参数$k+\ell$仍然是W[2]-难的。相比之下，Gupta等人（SAGT 2021）给出了路径图上关于$k$的FPT算法。为补充我们的结果并填补这一空白，我们提出了一种算法，当$\ell$为固定常数时，该算法在$k$上具有固定参数可解性（FPT）。