Geo-Graphs: An Efficient Model for Enforcing Contiguity and Hole Constraints in Planar Graph Partitioning

Authors:
Douglas M. King;Sheldon H. Jacobson;Edward C. Sewell;Wendy K. Tam Cho
Affiliations:
Department of Industrial and Enterprise Systems Engineering, University of Illinois, Urbana, Illinois 61801;Department of Computer Science, University of Illinois, Urbana, Illinois 61801;Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois 62026;Department of Political Science and Statistics, National Center for Supercomputing Applications, University of Illinois, Urbana, Illinois 61801
Venue:
Operations Research
Year:
2012

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Abstract

Political districting is an intractable problem with significant ramifications for political representation. Districts often are required to satisfy some legal constraints, but these typically are not very restrictive, allowing decision makers to influence the composition of these districts without violating relevant laws. For example, while districts must often comprise a single contiguous area, a vast collection of acceptable solutions i.e., sets of districts remains. Choosing the best set of districts from this collection can be treated as a planar graph partitioning problem. When districts must be contiguous, successfully solving this problem requires an efficient computational method for evaluating contiguity constraints; common methods for assessing contiguity can require significant computation as the problem size grows. This paper introduces the geo-graph, a new graph model that ameliorates the computational burdens associated with enforcing contiguity constraints in planar graph partitioning when each vertex corresponds to a particular region of the plane. Through planar graph duality, the geo-graph provides a scale-invariant method for enforcing contiguity constraints in local search. Furthermore, geo-graphs allow district holes which typically are considered undesirable to be rigorously and efficiently integrated into the partitioning process.