A tabu search heuristic and adaptive memory procedure for political districting

Abstract

In political districting problems, the aim is to partition a territory into electoral constituencies, subject to some side constraints. The most common side constraints include contiguity, population equality, compactness, and socio-economic homogeneity. We propose a formulation in which the various constraints are integrated into a single multicriteria function. We solve the problem by means of a tabu search and adaptive memory heuristic. The procedure is illustrated on real data from the city of Edmonton.

Introduction

In districting problems, the aim is to partition a territory into districts, subject to some side constraints. Typical districting problems include the drawing of political constituencies, school board boundaries, sales or delivery regions. Here we focus our attention on political districting. This problem is particularly important in democracies where each district elects a single member to a parliamentary assembly. One important issue at stake is equity (or population equality), i.e., all districts should have approximately the same number of voters in order to respect the “one-man, one-vote” principle. Furthermore, political districts must not be seen as favoring a particular political party. A famous case arose in Massachusetts in the early 19th century when the state legislature proposed a salamander-shaped district in order to gain electoral advantage. The governor of the state at that time was Elbridge Gerry and this practice became known as gerrymandering. Interesting accounts of gerrymandering cases are provided by Cain (1984) and Lewyn (1993). For a recent book on political districting the reader is referred to Grilli di Cortona et al. (1999).

To prevent political interference in the districting process, several states have set up a neutral commission whose functions include the drawing up of political boundaries satisfying a number of legislative and common sense criteria. Some criteria define the feasibility of a solution. For example, several legislatures impose that districts should be contiguous and that their (voting) population should fall within some interval. Other criteria help to assess the quality of a solution. An important criterion is that districts should be compact, the idea being to prevent the formation of odd-shaped districts that raise suspicions of gerrymandering. Other commonly used criteria are:

• the respect of natural boundaries, such as major bodies of water;

• the respect of some existing administrative or political subdivisions, like census tracts, townships;

• socio-economic homogeneity to ensure a better representation of residents who share common concerns or views;

• similarity to the existing plan so that an incumbent runs again in a similar district;

• the respect of integrity of communities, i.e., avoiding splitting some communities between several districts;

• equal probability of representation to ensure that some important minority groups have their fair share of representatives.

Some of these criteria can be disputed. For example, it may be argued that it is just as desirable to achieve socio-economic heterogeneity as it is important to reach homogeneity. Also the last criterion, and the reverse consisting of diluting the strength of any particular group, is itself a form of gerrymandering and should be handled with care. Similarly, any criterion aimed at providing a fair representation of political parties or at protecting safe seats may also be perceived as suspicious. For this reason, it may be argued that political data should not be used when designing districts. In the same spirit, more confidence is likely to be put in the process if it is computer-based, but even then some human intervention is required for the selection of criteria and for the determination of their relative weights.

The scientific literature on districting shows that there is no consensus on which criteria are legitimate and on how these should be measured (Williams, 1995). In this study, we will consider some of the most commonly accepted criteria: respect of major natural boundaries, contiguity, population equality, compactness, socio-economic homogeneity, similarity with the existing plan, and integrity of communities. This list should not, however, be viewed as limitative since our method can in principle work with any number of criteria as long as these can be measured. We propose a model that assigns weights to the various criteria, and a flexible solution methodology capable of producing high quality districting plans with respect to a set of weights. By altering weights or making several runs of the algorithm, decision makers should be able to generate a variety of solutions that will appeal to diverse interests.

Since the early 1960s, several heuristics have been proposed for the districting problem. All attempt to combine indivisible basic units such as census tracts or enumeration areas into feasible districts. These methods are based on one of the two integer linear programming formulations of the problem.

The first mathematical programming approach was proposed by Hess et al. (1965). It formulates the problem as an assignment problem with side constraints. Let I be the set of all basic units, and let J be the set of basic units used as potential district “seeds”. The cost c_ij of assigning unit i to seed j is a function of the Euclidean distance between the center of j and the center of i (typically c_ij is the square of that distance). The number of districts to be created is given and equal to m. The population of unit i is equal to p_i and the population of any district must lie within an interval [a,b]. Let x_ij be a binary variable equal to 1 if and only if unit i is assigned to seed j. The problem is then modeled as a capacitated m-median problem as follows:

$$\text{(}\text{F1}\text{)}\text{minimize}\text{∑}{}_{\mathrm{i\in I}}\text{∑}{}_{\mathrm{j\in J}}\text{c}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{ij}}$$$$\text{subject}\text{to}\text{∑}{}_{\mathrm{j\in J}}\text{x}{}_{\mathrm{ij}}\text{=1}\text{(i∈I),}$$$$\text{∑}{}_{\mathrm{j\in J}}\text{x}{}_{\mathrm{jj}}\text{=m,}$$$$\text{x}{}_{\mathrm{ij}}\text{⩽x}{}_{\mathrm{jj}}\text{(i∈I,j∈J),}$$$$\text{a⩽∑}{}_{\mathrm{i\in I}}\text{p}{}_{\mathrm{i}}\text{x}{}_{\mathrm{ij}}\text{⩽b}\text{(j∈J),}$$$$\text{x}{}_{\mathrm{ij}}\text{=0}\text{or}\text{1}\text{(i∈I,j∈J).}$$

In this formulation, the objective function measures compactness, while population equity is taken into account by constraints (4). Constraints (1) ensure that each basic unit is assigned to one district, and the number of districts is equal to m by constraint (2). By constraints (3), no basic unit can be assigned to an unselected seed. Note that these constraints are left out of the Hess et al. model. There is no guarantee that this formulation will produce contiguous districts, although this will be favored by the objective function. Thus solving (F1) to optimality does not in general yield a suitable solution to the districting problem, but it can produce an embryonic infeasible solution that can then be patched through local search.

This formulation can also serve as a guide for a heuristic. Given a reasonable set of m seed units, the remaining units can be iteratively assigned to these in a greedy fashion while ensuring that constraints (4) are satisfied. Again, it may be impossible to achieve contiguity. Since this type of approach does not require a linear objective, it is relatively easy to incorporate several terms into the objective, including constraints (4), each corresponding to a criterion to be satisfied. This is essentially the approach taken by most authors in the field, with various degrees of sophistication and a large number of variants (see, e.g., Vickrey, 1961; Weaver and Hess, 1963; Hess et al., 1965; Kaiser, 1966; Nagel, 1965, Nagel, 1972; Thoreson and Liittschwager, 1967; Morrill, 1973, Morrill, 1976; Bourjolly et al., 1981; Plane, 1982; Fleischmann and Paraschis, 1988; Browdy, 1990; Macmillan and Pierce, 1992). Several of these methods include a local search post-optimization phase consisting of moving basic units to adjacent districts, or swapping units between adjacent districts. The most sophisticated of these schemes, by Browdy (1990) and Macmillan and Pierce (1992), is based on simulated annealing. Local search is not only useful for improving the objective function, but it may also help the process move from an infeasible to a feasible solution.

In the second mathematical programming formulation, I is again the set of basic units and J is the set of all feasible districts. A binary coefficient a_ij is equal to 1 if and only if unit i belongs to district j. A cost c_j is assigned to district j and the number of districts is again equal to m. Binary variable x_j takes the value 1 if and only if district j is selected. The formulation, first proposed by Garfinkel and Nemhauser (1970) is

$$\text{(}\text{F2}\text{)}\text{minimize}\text{∑}{}_{\mathrm{j\in J}}\text{c}{}_{\mathrm{j}}\text{x}{}_{\mathrm{j}}$$$$\text{subject}\text{to}\text{∑}{}_{\mathrm{j\in J}}\text{a}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{j}}\text{=1}\text{(i∈I),}$$$$\text{∑}{}_{\mathrm{j\in J}}\text{x}{}_{\mathrm{j}}\text{=m,}$$$$\text{x}{}_{\mathrm{j}}\text{=0}\text{or}\text{1}\text{(j∈J).}$$

Since the number of potential districts is astronomical in most situations, Garfinkel and Nemhauser (1970) propose obtaining a good solution to (F2) by means of a truncated branch-and-bound tree. Recently, Mehrotra et al. (1998) have developed a column generation algorithm for this model. Their method remains a heuristic since the subproblem in which new columns are generated is NP-hard and is not solved optimally.

The purpose of this paper is to develop a tabu search (TS) algorithm for the political districting problem, an approach that has been highly successful for the solution of a host of combinatorial optimization problems (see, e.g., Glover and Laguna, 1997). Tabu search is advantageous in that it does not require a sophisticated integer linear programming apparatus, which makes its adoption by end-users easier. In addition to the standard TS methodology, we embed our algorithm within an adaptive memory procedure by Rochat and Taillard (1995).

The rest of this paper is organized as follows. In Section 2 the various criteria constituting the objective function of the model are formalized and cast in mathematical terms. The TS algorithm itself is described in Section 3. This is followed by computational results in Section 4, and by the conclusions in Section 5.