Gerrymandering on Graphs: Computational Complexity and Parameterized Algorithms

This paper studies gerrymandering on graphs from a computational viewpoint (introduced by Cohen-Zemach et al. [AAMAS 2018] and continued Ito 2019]). Our contributions are two-fold: conceptual computational. We propose generalization of the model studied al., where input consists graph n vertices representing set voters, m candidates \(\mathcal {C}\), weight function \(w_v: \mathcal {C}\rightarrow {\mathbb Z}^+\) for each voter \(v\in V(G)\) preference over candidates, distinguished candidate \(p\in positive integer k. The objective is to decide if it possible partition vertex into k districts (i.e., pairwise disjoint connected sets) such that p wins more than any other candidate. There several natural parameters associated with problem: number (k), voters (n), (m). problem known be NP-complete even \(k=2\), \(m=2\), G either complete bipartite (in fact \(K_{2,n}\), i.e., partitions size 2 n) or graph. Moreover, recently we Bentert [WG 2021], independently, showed NP-hard paths. means search FPT algorithms needs focus parameter n, subclasses forest (as family can transformed deleting one vertex). Circumventing these intractability results successfully obtain following algorithmic results.