Gerrymandering on graphs: Computational complexity and parameterized algorithms

Abstract

The practice of partitioning a region into areas to favor a particular candidate or a party in an election has been known to exist for the last two centuries. This practice is commonly known as gerrymandering. Recently, the problem has also attracted a lot of attention from complexity theory perspective. In particular, Cohen-Zemach et al. [AAMAS 2018] proposed a graph theoretic version of gerrymandering problem and initiated an algorithmic study around this, which was continued by Ito et al. [AAMAS 2019]. In this paper we continue this line of investigation and resolve an open problem in the literature, as well as move the algorithmic frontier forward by studying this problem in the realm of parameterized complexity.Our contributions in this article are two-fold, conceptual and computational. We first resolve the open question posed by Ito et al. [AAMAS 2019] about the computational complexity of gerrymandering when the input graph is a path. Next, we propose a generalization of the model studied in [AAMAS 2019], where the input consists of a graph on n vertices representing the set of voters, a set of m candidates C, a weight function w v : C → Z + for each voter v ∈ V (G) representing the preference of the voter over the candidates, a distinguished candidate p ∈ C, and a positive integer k. The objective is to decide if it is possible to partition the vertex set into k districts (i.e., pairwise disjoint connected sets) such that the candidate p wins more districts than any other candidate. There are several natural parameters associated with the problem: the number of districts the vertex set needs to be partitioned (k), the number of voters (n), and the number of candidates (m). The problem is known to be NP-complete even if k = 2, m = 2, and G is either a complete bipartite graph (in fact K 2,n , a complete bipartite graphs with one side of size 2 and the other of size n) or a complete graph. This hardness result implies that we cannot hope to have an algorithm with running time (n + m) f (k,m) let alone f (k, m)(n + m) O(1) , where f is a function depending only on k and m, as this would imply that P=NP. This means that in search for FPT algorithms we need to either focus on the parameter n, or subclasses of forest (as the problem is NP-complete on K 2,n , a family of graphs that can be transformed into a forest by deleting one vertex). Circumventing these intractable results, we successfully obtain the following algorithmic results.