Gerrymandering Planar Graphs
This addresses the problem of electoral district design for political scientists and computer scientists, offering theoretical insights into algorithmic feasibility and limitations, though it is incremental as it builds on prior work on simpler graph structures.
The paper tackles the computational complexity of gerrymandering in planar graphs, showing that the problem is NP-complete for general planar graphs with two candidates but solvable in polynomial time for λ-outerplanar graphs under certain constraints, and it provides a constant-factor approximation algorithm for unweighted planar graphs when the optimal value is sufficiently large.
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.