Multi-objective genetic algorithm with variable neighbourhood search for the electoral redistricting problem
Introduction
The redistricting problem (also known as zone design problem) is the process of dividing a geographic space or region of spatial units often represented as polygons into smaller sub-regions or districts (called clusters in this paper). Probably the most well-known instance of the zone design problem is the electoral redistricting problem. As reported in [1], electoral redistricting consists of the partitioning of area units, generally administrative units, into a predetermined number of zones (clusters) such that:
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the units in each zone are contiguous,
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each zone is geographically compact,
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and the sum of the populations of the area units is as similar as possible among all the clusters or lies within a predetermined range [2].
One of the main principles in the electoral redistricting problem is that each cluster should have approximately the same number of voters. In fact, each cluster will elect the same number of Parliament members. Additionally, each cluster's geometric shape should not be perceived as favouring a particular political party. Therefore, beside population equality and clusters’ contiguity, some other constraints can be added to this optimization problem. These constraints are, for instance, compactness, socio-economic homogeneity, and political equal probability of representation [2]. Although it is possible to add as many criteria as one may wish, literature reports that some of them may be questionable [2]. Thus, there is a current and general trend to consider that a minimum number of criteria will be simpler to understand by users and the final redistricting will be less suspicious. According to [2], the use of computer-based models will give users more confidence in the solution, even though in some cases some human intervention may be required in the selection of criteria and in the definition of their respective weights. However, it is also a generally accepted notion that human intervention should be as limited as possible, in order to ensure impartiality.
In this paper we propose a new method for the electoral redistricting problem based on multi-objective genetic algorithms (MOGA). Being strongly based on the idea of Pareto-front optimization, compared to the existing redistricting algorithms, the method we propose has the advantage of allowing the use of multi-criteria without the need of manually defining the weight of each criterion. Furthermore, the proposed algorithm presents several interesting novel features, amongst which are:
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an ad-hoc initialization method, that produces solutions that respect the constraints of the problem. Furthermore, this method is able to produce better initial solutions with respect to traditional random initialization methods of genetic algorithms (Section 4.4);
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particular genetic operators that ensure that the newly created solutions satisfy the contiguity constraint (Section 4.1);
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the integration (inside a multi-objective genetic algorithm (discussed in Section 4.2)) of a variable neighbourhood search (discussed in Section 4.5). This results in an effective exploration/exploitation ratio for the considered problem, thereby improving the quality of the final solutions.
Unlike other genetic algorithm based techniques, discussed in [1], thanks to the genetic operators we defined, the proposed algorithm needs only to check local contiguity (that is the contiguity of a set of area units composed of an area unit u and the area units that bound u) in order to ensure the global contiguity of the redistricting plan. This implies a considerable reduction in the running time of the algorithm.
We applied the proposed method to five different US states, comparing the results obtained with those returned by several different state-of-the-art techniques presented in [3]. Section 2 presents a definition of the redistricting problem, along with some of the constraints used. Section 3 contains a review of some of the most well-known existing redistricting studies and techniques. In Section 4 a detailed description of the proposed method is provided. Section 5 presents the experimental settings and the results obtained for the five US states considered. Finally, Section 6 concludes and suggests ideas for possible future research.
Section snippets
Redistricting problem: definition and constraints
As mentioned above, in redistricting problems the aim is to aggregate n geo-spatial regions (also called zones or area units) into c partitions (also called districts or clusters), subject to some constraints. Electoral clusters should be spatially continuous, as compact as possible, and all of them should have approximately the same number of inhabitants [2]. Many other objectives can also be considered. For instance, respecting, as much as possible, natural or administrative boundaries;
Related work
To solve redistricting problems, several approaches have been proposed so far. In this section we discuss several existing works concerning the redistricting problem. We focus on the techniques that we have used for comparison with our proposed method. These techniques are graph partitioning, simulated annealing, genetic algorithm, and the so called Constrained Polygonal Spatial Clustering algorithm.
An exhaustive presentation of all the existing works goes beyond the objectives of this paper.
The proposed method
In this section we describe the proposed redistricting method. It is a hybrid algorithm that combines the NSGA-II technique (used to optimize both the objective functions presented in Section 2) with a variable neighbourhood search algorithm. All the components of the algorithm are described in the following sections and, finally, the resulting combined method is presented.
Experimental study
In this section we evaluate the performance of the proposed redistricting algorithm. We compare it with the graph partitioning, SARA, GA, and CPSC algorithms described in Section 3. We performed the experimental study on five redistricting test problems:
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For the first redistricting problem we used the census tract data set (year 2000) for the state of Nebraska. The total number of polygons in Nebraska is 505. The state of Nebraska has been assigned three seats in the US congress. Thus, the
Conclusions and future work
In this paper we proposed a multi-objective genetic algorithm with a variable neighbourhood search for addressing the electoral redistricting problem. The experimental study we presented, performed on five “real world” instances of this problem, indicates the appropriateness of the new algorithm for producing redistricting plans. The proposed algorithm was consistently able to produce redistricting plans characterized by better compactness and better population equality with respect to other
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