Gerrymandering Planar Graphs

TL;DR

The paper analyzes the computational complexity of map redistricting (gerrymandering) on planar graphs, formalizing partitions into $k$ connected districts and aiming to maximize blue-winning districts with vertex weights and candidate approvals. It introduces a dynamic-programming framework for graphs of bounded treewidth (and thus $\lambda$-outerplanar graphs) to solve the problem exactly when the number of candidates is constant and weights are polynomially bounded, and it establishes strong hardness and inapproximability results for general planar graphs with varying candidate counts. It also provides a constant-factor approximation for unweighted planar graphs when the optimal number of winning districts is sufficiently large, and a PTAS for the singleton-winning district variant using Baker's technique, linking singleton-wins to the broader problem. Collectively, these results delineate the tractability frontier on planar graphs, inform algorithmic directions for redistricting, and highlight open questions about PTAS applicability to the general problem and equal-sized districts.

Abstract

We study the computational complexity of the map redistricting problem (gerrymandering). Mathematically, the electoral district designer (gerrymanderer) attempts to partition a weighted graph into $k$ connected components (districts) such that its candidate (party) wins as many districts as possible. Prior work has principally concerned the special cases where the graph is a path or a tree. Our focus concerns the realistic case where the graph is planar. We prove that the gerrymandering problem is solvable in polynomial time in $λ$-outerplanar graphs, when the number of candidates and $λ$ are constants and the vertex weights (voting weights) are polynomially bounded. In contrast, the problem is NP-complete in general planar graphs even with just two candidates. This motivates the study of approximation algorithms for gerrymandering planar graphs. However, when the number of candidates is large, we prove it is hard to distinguish between instances where the gerrymanderer cannot win a single district and instances where the gerrymanderer can win at least one district. This immediately implies that the redistricting problem is inapproximable in polynomial time in planar graphs, unless P=NP. This conclusion appears terminal for the design of good approximation algorithms -- but it is not. The inapproximability bound can be circumvented as it only applies when the maximum number of districts the gerrymanderer can win is extremely small, say one. Indeed, for a fixed number of candidates, our main result is that there is a constant factor approximation algorithm for redistricting unweighted planar graphs, provided the optimal value is a large enough constant.

Figures (1)

• Figure 2: The Cut and Connect procedure

Theorems & Definitions (20)

• theorem 1
• theorem 2
• lemma 1
• theorem 3
• theorem 4
• corollary 1
• theorem 5
• lemma 2
• lemma 3
• lemma 4