19 research outputs found

    A Graph Partitioning Model of Congressional Redistricting

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    Redrawing congressional districts in the United States is a constitutionally required, yet politically controversial, task undertaken after each decennial census. Federal law requires contiguous, `relatively compact\u27 congressional districts that maintain `approximately equal\u27 population. Controversy is introduced when individual states redraw their districts, or redistrict, using partisan committees. States such as Ohio continue to redistrict with a committee appointed according to the current proportion of legislators\u27 political parties to the whole. When political parties have majority power in redistricting committees, they can draw districts in a way that gives their party the best chance to keep its majority representation, a process called gerrymandering. Mathematical redistricting models seek an unbiased computational approach to the problem. Rather than trust partisan committees, mathematical modeling approaches rely upon well-defined methods in computational geometry, graph theory, game theory, and other fields. Here, we discuss two such approaches. The first, given as a background for comparison, constructs Voronoi diagrams to redistrict states into convex polygons, which are generally considered `compact\u27. We give greater emphasis to a new model that discretizes a state\u27s population and partitions it into regions of approximately equal population. This model, our main focus, relies upon graph partitioning to achieve the desired result and uses census population data as the sole parameter in redistricting

    On the Computational Tractability of a Geographic Clustering Problem Arising in Redistricting

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    Redistricting is the problem of dividing up a state into a given number k of regions (called districts) where the voters in each district are to elect a representative. The three primary criteria are: that each district be connected, that the populations of the districts be equal (or nearly equal), and that the districts are "compact". There are multiple competing definitions of compactness, usually minimizing some quantity. One measure that has been recently been used is number of cut edges. In this formulation of redistricting, one is given atomic regions out of which each district must be built (e.g., in the U.S., census blocks). The populations of the atomic regions are given. Consider the graph with one vertex per atomic region and an edge between atomic regions with a shared boundary of positive length. Define the weight of a vertex to be the population of the corresponding region. A districting plan is a partition of vertices into k pieces so that the parts have nearly equal weights and each part is connected. The districts are considered compact to the extent that the plan minimizes the number of edges crossing between different parts. There are two natural computational problems: find the most compact districting plan, and sample districting plans (possibly under a compactness constraint) uniformly at random. Both problems are NP-hard so we consider restricting the input graph to have branchwidth at most w. (A planar graph’s branchwidth is bounded, for example, by its diameter.) If both k and w are bounded by constants, the problems are solvable in polynomial time. In this paper, we give lower and upper bounds that characterize the complexity of these problems in terms of parameters k and w. For simplicity of notation, assume that each vertex has unit weight. We would ideally like algorithms whose running times are of the form O(f(k,w) n^c) for some constant c independent of k and w (in which case the problems are said to be fixed-parameter tractable with respect to those parameters). We show that, under standard complexity-theoretic assumptions, no such algorithms exist. However, the problems are fixed-parameter tractable with respect to each of these parameters individually: there exist algorithms with running times of the form O(f(k) n^{O(w)}) and O(f(w) n^{k+1}). The first result was previously known. The new one, however, is more relevant to the application to redistricting, at least for coarse instances. Indeed, we have implemented a version of the algorithm and have used to successfully find optimally compact solutions to all redistricting instances for France (except Paris, which operates under different rules) under various population-balance constraints. For these instances, the values for w are modest and the values for k are very small
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