Multi-objective genetic algorithm with variable neighbourhood search for the electoral redistricting problem

https://doi.org/10.1016/j.swevo.2017.04.003Get rights and content

Abstract

In a political redistricting problem, the aim is to partition a territory into electoral districts or clusters, subject to some constraints. The most common of these constraints include contiguity, population equality, and compactness. We propose an algorithm to address this problem based on multi-objective optimization. The hybrid algorithm we propose combines the use of the well-known Pareto-based NSGA-II technique with a novel variable neighbourhood search strategy. A new ad-hoc initialization method is also proposed. Finally, new specific genetic operators that ensure the compliance of the contiguity constraint are introduced. The experimental results we present, which are performed considering five US states, clearly show the appropriateness of the proposed hybrid algorithm for the redistricting problem. We give evidence of the fact that our method produces better and more reliable solutions with respect to those returned by the state-of-the-art methods.

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:

  • the units in each zone are contiguous,

  • each zone is geographically compact,

  • 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:

  • 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);

  • particular genetic operators that ensure that the newly created solutions satisfy the contiguity constraint (Section 4.1);

  • 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:

  • 1.

    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

References (45)

  • M. Altman

    Is automation the answer: the computational complexity of automated redistricting

    Rutgers Comput. Law Technol. J.

    (1997)
  • D.J. Rossiter et al.

    Program group: the identification of all possible solutions to a constituency-delimitation problem

    Environ. Plan. A

    (1981)
  • M. Keane

    The size of the region-building problem

    Environ. Plan. A

    (1975)
  • Zhenping Li, Rui-Sheng Wang, Yong Wang, A quadratic programming model for political districting problem, The First...
  • Juan Carlos Duque et al.

    Supervised regionalization methods: a survey

    Int. Reg. Sci. Rev.

    (2007)
  • Jorg Kalcsics et al.

    Towards a unified territorial design approach - applications, algorithms and GIS integration

    Top

    (2005)
  • Federica Ricca et al.

    Political districting: from classical models to recent approaches

    Ann. Oper. Res.

    (2013)
  • Andris A. Zoltners et al.

    Sales territory design: thirty years of modeling and implementation

    Mark. Sci.

    (2005)
  • William Vickrey

    On the prevention of gerrymandering

    Political Sci. Q.

    (1961)
  • L.D. Bodin

    A districting experiment with a clustering algorithm

    Ann. New Y. Acad. Sci.

    (1973)
  • James B. Weaver et al.

    A procedure for nonpartisan districting: development of computer techniques

    Yale Law J.

    (1963)
  • S.W. Hess et al.

    Nonpartisan political redistricting by computer

    Oper. Res.

    (1965)
  • Cited by (33)

    • Mixed integer linear programming model and an effective algorithm for the bi-objective double-floor corridor allocation problem

      2021, Computers and Operations Research
      Citation Excerpt :

      This technique is utilized to improve the local search capability by adaptively transforming between a deep-searching strategy and a broad-searching strategy. As far as we know, in current literature, the GAVNS has been widely applied to various domains and their excellent performance has been evaluated, such as the electoral redistricting problem (Vanneschi et al., 2017), parallel machine scheduling (Alharkan et al., 2018) and vehicle routing problem with multiple time windows (Belhaiza et al., 2017). Afterwards, to evaluate our proposed GAVNS, numerous instances of sizes varying from 60 to 300 were utilised to make several numerical experiments, and the comparison between our results and those of other algorithms in the current literature have confirmed the performance of our GAVNS.

    • A modified equilibrium optimizer using opposition-based learning and novel update rules

      2021, Expert Systems with Applications
      Citation Excerpt :

      As technology has developed, increasingly complex optimization problems have continually emerged. However, traditional mathematical optimization methods are too inefficient and inaccurate to meet the needs of many current problems (Hu et al., 2020; Sánchez et al., 2020; Vanneschi et al., 2017). Therefore, the use of metaheuristic algorithms to solve optimization problems has attracted the attention of a large number of researchers (Bernal et al., 2020; Ghahremani-Nahr et al., 2019; Li et al., 2018).

    • Algorithms for gerrymandering over graphs

      2021, Theoretical Computer Science
    View all citing articles on Scopus
    View full text