Gerrymandering on Graphs: Computational Complexity and Parameterized Algorithms

Abstract

This paper studies gerrymandering on graphs from a computational viewpoint (introduced by Cohen-Zemach et al. [AAMAS 2018] and continued by Ito et al. [AAMAS 2019]). Our contributions are two-fold: conceptual and computational. We propose a generalization of the model studied by Ito et al., where the input consists of a graph on n vertices representing the set of voters, a set of m candidates $\mathcal {C}$, a weight function $w_v: \mathcal {C}\rightarrow {\mathbb Z}^+$ for each voter $v\in V(G)$ representing the preference of the voter over the candidates, a distinguished candidate $p\in \mathcal {C}$, 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 (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 $K_{2,n}$, i.e., partitions of size 2 and n) or a complete graph. Moreover, recently we and Bentert et al. [WG 2021], independently, showed that the problem is NP-hard for paths. This means that the search for FPT algorithms needs to focus either 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 intractability results we successfully obtain the following algorithmic results.

• A $2^n (n+m)^{\mathcal {O}(1)}$ time algorithm on general graphs.

• FPT algorithm with respect to k (an algorithm with running time $2^{{\mathcal O}(k)}n^{{\mathcal O}(1)}$) on paths in both deterministic and randomized settings, even for arbitrary weight functions. Whether the problem is FPT parameterized by k on trees remains an interesting open problem.

Our algorithmic results use sophisticated technical tools such as representative set family and Fast Fourier Transform based polynomial multiplication, and their (possibly first) application to problems arising in social choice theory and/or algorithmic game theory is likely of independent interest to the community.

Sushmita Gupta supported by SERB-Starting Research Grant (SRG/2019/001870). Fahad Panolan supported by Seed grant, IIT Hyderabad (SG/IITH/F224/2020-21/SG-79). Sanjukta Roy supported by the WWTF research grant (VRG18-012). Saket Saurabh supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.