A partial differential equations approach to defeating partisan gerrymandering

Abstract

We introduce a novel partial differential equations approach for addressing the problem of partisan gerrymandering. Our method is based on volume preserving curvature flow, a partial differential equation which we adapt to smooth voting district boundaries while preserving equal voting populations. We show that every step of the flow minimizes a "compactness energy", allowing us to demonstrate that our method produces more "compact" and reasonable district maps. We compute the flow using a variant of "auction dynamics" --- an efficient MBO type algorithm for computing volume preserving curvature flows. This "auction dynamics" approach can be used to generate hundreds of reasonable maps in a matter of seconds without parallelization. The compactness energy provides a way of comparing proposed districtings of a given state. We demonstrate both the map generation and map comparison features of our approach for several different states.