You can have your cake and redistrict it too

The design of algorithms for political redistricting generally takes one of two approaches: optimize an objective such as compactness or, drawing on fair division, construct a protocol whose outcomes guarantee partisan fairness. We aim to have the best of both worlds by optimizing an objective subject to a binary fairness constraint. As the fairness constraint we adopt the geometric target, which requires the number of seats won by each party to be at least the average (rounded down) of its outcomes under the worst and best partitions of the state; but we extend this notion to allow the two parties to compute their targets with respect to different election datasets. Our theoretical contribution is twofold: we introduce a new model of redistricting that closely mirrors the classic model of cake-cutting and we prove the feasibility of the geometric target in this model. Our empirical results, which use real election data and maps of six US states, demonstrate that the geometric target is feasible in practice and that imposing it as a fairness constraint comes at almost no cost to three well-studied optimization objectives.First author draf

A cutting-plane method for contiguity-constrained spatial aggregation

Aggregating areas into larger regions is a common problem in spatial planning, geographic information science, and cartography. The aim can be to group administrative areal units into electoral districts or sales territories, in which case the problem is known as districting. In other cases, area aggregation is seen as a generalization or visualization task, which aims to reveal spatial patterns in geographic data. Despite these different motivations, the heart of the problem is the same: given a planar partition, one wants to aggregate several elements of this partition to regions. These often must have or exceed a particular size, be homogeneous with respect to some attribute, contiguous, and geometrically compact. Even simple problem variants are known to be NP-hard, meaning that there is no reasonable hope for an efficient exact algorithm. Nevertheless, the problem has been attacked with heuristic and exact methods. In this article we present a new exact method for area aggregation and compare it with a state-of-the-art method for the same problem. Our method results in a substantial decrease of the running time and, in particular, allowed us to solve certain instances that the existing method could not solve within five days. Both our new method and the existing method use integer linear programming, which allows existing problem solvers to be applied. Other than the existing method, however, our method employs a cutting-plane method, which is an advanced constraint-handling approach. We discuss this approach in detail and present its application to the aggregation of areas in choropleth maps

A cutting-plane method for contiguity-constrained spatial aggregation

Aggregating areas into larger regions is a common problem in spatial planning, geographic information science, and cartography. The aim can be to group administrative areal units into electoral districts or sales territories, in which case the problem is known as districting. In other cases, area aggregation is seen as a generalization or visualization task, which aims to reveal spatial patterns in geographic data. Despite these different motivations, the heart of the problem is the same: given a planar partition, one wants to aggregate several elements of this partition to regions. These often must have or exceed a particular size, be homogeneous with respect to some attribute, contiguous, and geometrically compact. Even simple problem variants are known to be NP-hard, meaning that there is no reasonable hope for an efficient exact algorithm. Nevertheless, the problem has been attacked with heuristic and exact methods. In this article we present a new exact method for area aggregation and compare it with a state-of-the-art method for the same problem. Our method results in a substantial decrease of the running time and, in particular, allowed us to solve certain instances that the existing method could not solve within five days. Both our new method and the existing method use integer linear programming, which allows existing problem solvers to be applied. Other than the existing method, however, our method employs a cutting-plane method, which is an advanced constraint-handling approach. We discuss this approach in detail and present its application to the aggregation of areas in choropleth maps

THEORY AND ALGORITHMS FOR COMMUNITY DETECTION AND CLUSTERING IN NETWORKS

This dissertation is focused on certain clustering and partitioning problems in networks. We present a comprehensive study of the maximum independent union of cliques problem and its generalizations in uniform random graphs. The main result is the logarithmic upper bound, similarly to the maximum clique, which suggests a subexponential algorithm. Then we propose a parallel version of Russian Doll Search, an algorithm that can be used to find the maximum independent union of cliques. We enhance existing verification procedure for this problem by a simple observation, which also leads to an elegant constant time biclique verification. Finally, we perform the first computational study for finding Hadwiger’s number of a graph. We present several integer formulations, scale-reduction techniques, heuristics, and bounds, together with a scheme for future exact combinatorial algorithms