Geospatial Analytics to Support District Efficiency, Fairness and Equity ^†

Abstract

Districting is essential in establishing efficient, fair, and equitable boundaries across various domains such as trade and service provision, land-use development, school attendance, and political representation. This paper introduces a robust spatial optimization model tailored to delineate a districting strategy that accounts for issues of equinumerosity, compactness, and contiguity, among others. The model is designed to ensure an even distribution of population across districts while ensuring compact and contiguous boundaries. By integrating advanced GIScience and spatial optimization techniques, the model aims to maximize compactness metrics and balance population, producing districts that are both geometrically efficient and demographically representative. Explicit contiguity constraints are imposed to guarantee geographical connectedness among districts. To assess the efficacy of the proposed methodology, application to a school district in California is considered. Our findings highlight GIScience capabilities to identify district boundaries that outperform traditional districting schemes in addressing population balance and compactness.

1. Introduction

Districting plays a critical role in various aspects of human lives, impacting education, business services, and political representation. For example, school districts determine which school children attend, influencing their education experience, the resources available, and the formation of neighborhood communities. Business often defines service areas to optimize delivery routes, allocate resources efficiently, and target marketing efforts. Good districting ensures access, but also has implications for customer service quality. In politics, the way congressional districts are drawn can affect the balance of power in legislative bodies, impacting policy decisions and representation. Given these profound effects, quality districting approaches are essential for establishing efficient, fair, and equitable boundaries across domains.

Key principles have been established for a fair political districting over centuries: equinumerosity, contiguity, and compactness ([1,2,3]; and so on). Firstly, districts must represent an equal number of people, known as equinumerosity. This principle is fundamental not only for electoral districting, but for delineating service areas, ensuring service availability and balancing facility utilization. Secondly, districts must be contiguous, meaning that all parts of a district are physically connected. Contiguity maintains the integrity and coherence of districts, avoiding fragmented and disjointed areas that could dilute representation. Lastly, districts should be as spatially compact as possible to avoid bizarre shapes that serve partisan interests. While compactness raises theoretical and methodological questions, it is crucial for preventing manipulation of district boundaries, a goal that cannot be achieved through contiguity alone.

Based on these principles, several studies have utilized optimization approaches to create/identify good districts [1,2,4,5,6,7]. It is only recently that solving districting problems with exact approaches, rather than heuristics, has become feasible given advances in computational capabilities. Still, problem size and complexity remain significant concerns. Many optimization approaches focus on one of the principles as the objective while addressing the others as constraints or implications, resulting in a single districting outcome. However, multi-objective approaches can identify Pareto-optimal results that address all of the principles simultaneously, offering decision makers a variety of options [8,9]. A bi-objective spatial optimization is proposed here addressing the three principles directly: to maximize spatial compactness and to balance the population simultaneously, with the contiguity constraint. The major challenge is due to the intricate complexity of addressing contiguity and compactness, and the large size of the districting problem, which complicates the application of Integer Linear Programming approaches especially under multi-objective considerations. This paper aims to bridge this gap in districting.

The remainder of this paper is structured as follows. Section 2 provides a review of the theoretical foundations underlying our approach. Section 3 describes the mathematical model formulated to address the proposed problem. Application results are presented in Section 4, followed by a discussion in Section 5, which includes the summary of findings and concluding remarks.

2. Background

The Apportionment Act of 1901 (repealed in 1929), along with numerous court decisions, mandates compactness for U.S. House districts. Additionally, 18 state constitutions and 37 states require their legislative districts to be compact [10]. Gerrymandering refers to any manipulation of district lines for partisan purposes. In Davis v. Bandemer (478 U.S. 109., 1986 [11]), the Supreme Court ruled that claims of gerrymandering are subject to judicial review as potential violations of the Equal Protection Clause of the Constitution. Compactness has come to be a key factor in districting to counter boundary manipulation for political gain. However, there are over a hundred practical definitions of compactness. Notable measures include Polsby and Popper (1991) [3], Schwartzberg (1965) [12], Reock (1961) [13], and Normalized Moment of Inertia (Li et al., 2013 [14]). These measures use geometric and mathematical attributes to determine the compactness of the shape, capturing different characteristics. The challenge is that these methods are non-linear and often quadratic, making them difficult to address using exact solution approaches. As a result, spatial compactness has been ignored or only implicitly reflected in districting optimization models [1,2,4,15,16,17].

Hess et al. (1965) [1] is considered the earliest operations research work in districting attempting to minimize population moment of inertia by penalizing the population-weighted squared distance between the spatial units within the same district. This approach is related to spatial compactness through its objective of minimizing population dispersion within districts. This approach has influenced later studies [4,18,19], but does not necessarily address overall spatial compactness across districts. Recent work by Almeida et al. (2022) [6] highlights an approach of minimizing the district boundary length, explicitly considering spatial compactness as an objective in an Integer Linear Programming model. The approach of minimizing district boundaries effectively addresses spatial compactness, as supported by Polsby and Popper (1991) [3].

Equinumerosity is generally addressed in districting models [1,2,4,18,20,21]. While some approaches [1,4,6] consider equinumerosity in constraints, other works [2,19] include it as an objective in their model. Equinumerosity can be approached using capacitated constraints. Various measures incorporate capacity balance into Linear Programming models [2,22,23,24,25]. Utility depends on the computational challenges, including the number of decision variables and problem complexity. In this paper, district assignment and contiguity constraint already introduce a huge number of decision variables; therefore, the minimization of maximum district population approach [23,26,27] is considered as it introduces the least amount of decision variables.

Contiguity constraints impose a large number of decision variables in an optimization model. Therefore, research has sought to enhance computational efficiency [28]. Oehrlein and Haunert (2017) [29] propose a flow-based formulation, adapted from [30]. Validi et al. (2021) [4] updated the approach with a new flow-based formulation avoiding the big-M constraints. The work of Murray and Church (2023) [31] is notable because it significantly reduces the number of needed decision variables. Almeida et al. (2022) [6] proposed two new contiguity formulations, shortest-path-based and tree-based, concluding that the tree-based formulation finds more solutions while the shortest-path-based constraints miss some solution configurations.

Multi-objective optimization approaches by Arcese et al. (1992) [32] and Ricca et al. (2008) [33] considered equinumerosity, compactness, and conformity to administrative boundaries, incorporating contiguity constraints. However, this work focused on population compactness, measuring population dispersion within a district. Moreover, heuristic approaches were utilized, such as clustering and local search, to solve the districting optimization problem instead of an exact approach [17,19,32,33,34,35]. Additionally, other districting studies have focused on optimizing traits such as service quality, accessibility, and minority representation [7,19,35]. However, a bi-objective spatial optimization problem that explicitly addresses the three principles of equinumerosity, spatial compactness, and contiguity has not yet been defined and solved.

3. Methods

This paper introduces a bi-objective spatial optimization problem for districting, extending the Almeida et al. (2022) [6] model. Consider the following notation:

$u$: the total number of spatial units to be districted;

$U$: the set of spatial units numbered from 1 to $u$;

$P_i$: Population at spatial unit $i$;

$N_i$: the set of adjacent neighbors of spatial unit $i\in U$;

$p$: the total number of districts;

$D$: the set of districts, numbered from 1 to $p$;

$L_{ij}$: the length of sharing border between adjacent spatial units $i, j\in U$;

$m$: the maximum number of spatial units that can be assigned to a district;

$M$: the set of possible depths in a tree from 1 to $m$.

Given a set of $u$ spatial units (entire set $U$), $p$ districts will be determined (entire set $D$) accounting for the adjacency between spatial units. The decision variables involved in the model are as follows:

$$X_{ki}=\left\{\begin{array}{c}1: \mathrm{if} \mathrm{spatial} \mathrm{unit} i\in U \mathrm{is} \mathrm{assigned} \mathrm{to} \mathrm{district} k\in D\\ 0: \mathrm{otherwise} \end{array}\right.$$

$$B_{kij}=\left\{\begin{array}{c}1: \mathrm{if} \mathrm{spatial} \mathrm{units} i, j\in U \mathrm{both} \mathrm{belong} \mathrm{to} \mathrm{district} k\in D\\ 0: \mathrm{otherwise} \end{array}\right.$$

$$P_{ij}=\left\{\begin{array}{c}1: \mathrm{if} \mathrm{spatial} \mathrm{unit} j\in N_i \cup \left\{0\right\} \mathrm{is} \mathrm{predecessor} \mathrm{of} i\in U\\ 0: \mathrm{otherwise} \end{array}\right.$$

$$H_{ih}=\left\{\begin{array}{c}1: \mathrm{if} \mathrm{spatial} \mathrm{unit} i\in U \mathrm{is} \mathrm{at} \mathrm{depth} \mathrm{h} \in \mathrm{M} \mathrm{in} \mathrm{the} \mathrm{tree}\\ 0: \mathrm{otherwise} \end{array}\right.$$

Let $X_{ki}$ and $B_{kij}$ represent decision variables to track the assignment of spatial units ($i \mathrm{or} j$) to district $k$. Additionally, $P_{ij}$ and $H_{ih}$ are employed to ensure tree-based contiguity. The mathematical model, which incorporates these decision variables to address the objectives of maximizing compactness and balancing population across districts, is as follows:

$$Maximize {\displaystyle \sum }_{k\in D}{\displaystyle \sum }_{i\in U}{\displaystyle \sum }_{j\in N_i}{L}_{ij}{B}_{kij}$$

(1)

$$Minimize \underset{k\in D}{\max }{\displaystyle \sum }_{i\in U}{P}_i{X}_{ki}$$

(2)

Objective (1) aims to maximize the shared boundary length within the districts. Given the finite set of spatial units, maximizing the internal shared boundary length will minimize the overall district boundary, thereby enhancing compactness [3]. Objective (2) seeks to minimize the maximum district population, thereby balancing the population across districts [23].

$${{\displaystyle \sum }}_{k\in D}{X}_{ki}=1 \forall i\in U$$

(3)

$${{\displaystyle \sum }}_{i\in U} X_{ki}\ge 1 \forall k\in D$$

(4)

$$X_{ki}-B_{kij}\ge 0 \forall k\in D, i\in U, j\in N_i$$

(5)

$$X_{kj}-B_{kij}\ge 0 \forall k\in D, i\in U, j\in N_i$$

(6)

$$X_{ki}, B_{kij}\in \left\{0,1\right\}$$

(7)

$$({{\displaystyle \sum }}_{j\in N_i}{P}_{ij})+P_{i0}=1 \forall i\in U$$

(8)

$$-X_{ki}+X_{kj}-P_{ji}\ge -1 \forall k\in D, i\in U, j\in N_i$$

(9)

$$P_{ij}+P_{ji}\le 1 \forall i\in U, j\in N_i$$

(10)

$$-X_{ki}-X_{kj}-P_{i0}-P_{j0}\ge -3 \forall k\in D, i,j\in U, i\ne j$$

(11)

$${{\displaystyle \sum }}_{h=1}^m{H}_{ih}=1 \forall i\in U$$

(12)

$$-P_{i0}+H_{i1}\ge 0 \forall i\in U$$

(13)

$$-P_{ij}-H_{jh}+H_{i\left(h+1\right)}\ge -1 \forall i\in U, j\in N_i, h\in M\left\{m\right\}$$

(14)

$$-H_{im}-P_{ji}\ge -1 \forall i\in U, j\in N_i$$

(15)

$${{\displaystyle \sum }}_{i\in U}{P}_{i0}=p$$

(16)

$$P_{ij}\in \left\{0,1\right\} \forall i\in U, j\in N_i\cup \left\{0\right\}$$

(17)

$$H_{kh}\in \left\{0,1\right\} \forall i\in U, h\in M$$

(18)

Constraints (3)–(7) define the assignment and relationships of spatial units within districts. Constraint (3) ensures that each spatial unit $i$ is assigned to only one district. Constraint (4) guarantees that each district will contain at least one spatial unit, thereby generating exactly $p$ districts as specified by the parameter. Constraints (5) and (6) ensure that $B_{kij}$ is set to one only when spatial units $i$ and $j$ are allocated to the same district k. Constraint (7) sets the $X_{ki}$ and $B_{kij}$ to be binary variables. Constraints (8)–(16) are tree-based contiguity constraints. Constraint (8) stipulates that for any spatial unit $i$, its predecessor in the district $k$ tree must either be a neighboring spatial unit or the root node (depth 0) of the tree. Constraint (9) mandates that neighboring spatial units $i$ and $j$ can have a predecessor relationship only when they belong to the same district. Constraint (10) ensures that the predecessor relationship is unidirectional. Constraint (11) requires that each district tree has a single root node. Constraint (12) specifies that each spatial unit can only be assigned one depth level, and constraint (13) dictates that the root node must always be at depth 1. Constraint (14) ensures that the depth of a spatial unit $i$ is exactly one level higher than its predecessor. Constraint (15) mandates that spatial units at depth $m$ must be leaf nodes, meaning they cannot be predecessors to other nodes. Finally, constraint (16) enforces that the number of root nodes equals the number of districts $p$.

The proposed model will generate a Pareto-optimal solution set, showing the balance between the two objectives, compactness and equinumerosity, while ensuring that districts are contiguous.

4. Application Results

The Hope Elementary School District in Santa Barbara County, California, includes three elementary schools. It is 14.36 square miles in size defined by 200 Census blocks, with a total population of 16,099 according to the 2020 Census. The blocks need to be configured into three school attendance zones. The proposed model (1)–(18) was implemented to ensure the school attendance zones are following the principles of districting: equinumerosity, contiguity, and compactness.

Analysis was carried out on a desktop personal computer, equipped with Intel^® Xeon^® W3-2423 CPU computer with 64 GB of RAM and an NVIDIA^® RTX™ A6000 GPU with 48 GB of VRAM. The proposed model (1)–(18) was implemented in Python and solved using Gurobi (v 11.0.0). Neighbors or adjacent units are those that share a common point or boundary, referred to as “queen” adjacency. To address the bi-objective optimization problem, the constraint method was utilized [8]. Objective (2) was converted into a constraint by setting the maximum district population to be less than a certain value, starting from the ideal number, ${\displaystyle \frac{1}{p}}\sum P_i=5366.33$, and incrementing by 100 until the total population, $\sum P_i=16,099$, was reached. It is important to note that the initial values for decision variables were strategically assigned. For the first instance of the constraint method, the K-means algorithm output was used to set the initial values, ensuring to start with contiguous districts, although not within the constrained objective bound. Subsequently, as the constrained objective bound increased in the iterative process, the previous solution was used as the initial value of decision variables in the next iteration. This facilitated an effective search within the solution space, considering the complexity of finding an initial feasible solution due to contiguity constraints and objective bound. A limit of one hour was imposed on each instance.

From the 117 constraint method instances, 35 unique solutions and 17 non-dominated solutions were identified, with an optimality gap ranging from 0 to 9%. The average optimality gap was 1.37%. Five of the initial instances with a constraint objective bound of 5800 or less were infeasible. Figure 1 summarizes the 35 solutions with current districting highlighted in blue and non-dominated solutions highlighted in red. The X-axis represents the shared boundary length in miles, while the Y-axis indicates the maximum district population. According to Figure 1, current districting is not Pareto-optimal, and is dominated by five feasible solutions. The leftmost non-dominated solution is an equinumerosity best solution. This solution has a shared boundary length of 171.35 miles and a maximum district population of 5837. Even though this solution provides less compact districts than current districting, it achieves a more balanced population across districts by reducing the maximum district population by 690. In contrast, the rightmost non-dominated solution is a compactness best solution that offers the most compact districts with a shared boundary length of 190.54 miles, but with an increased maximum district population of 10,432. Since the extreme solutions are superior in one objective but significantly inferior in another, a tradeoff solution can be highlighted. The second leftmost non-dominated solution has a shared boundary length of 183.34 miles, and maximum district population of 5840. The solution provides much more compact districts than the equinumerosity best solution, while the maximum district population only increases by 3. The tradeoff solution is more compact than current districting, and more balanced in terms of population.

Figure 2 displays the spatial configurations for the current districting and the highlighted solutions: equinumerosity best, compactness best, and tradeoff solutions. The district boundaries are illustrated, while each district is color-coded based on population. Figure 2a demonstrates the current districting, Figure 2b shows the equinumerosity best solution, Figure 2c presents the compactness best solution, and Figure 2d illustrates the tradeoff solution. Compared to Figure 2a, Figure 2b reveals more complex district boundaries but better population balance, as indicated by the more uniform colors across districts. The compactness best solution in Figure 2c shows that one district has only one block included, leading to poor population balance but compact district shapes. Therefore, the tradeoff solution in Figure 2d offers the most desirable outcome. The solution provides more compact boundaries than current districting, while achieving a more balanced population across districts.

5. Discussion and Conclusions

The strategy for identifying the complete Pareto-optimal set can be further discussed. In this study, a 100-interval increment was employed for the constraint method instead of a 1-interval increment. While a 1-interval increment would theoretically yield the complete set of Pareto-optimal solutions, it would also be computationally prohibitive due to the significant increase in required resources and time. The choice of a 100-interval increment is a practical compromise, balancing the need for solution quality with computational feasibility. This approach allows for the identification of a representative subset of Pareto-optimal solutions, though it may miss some finer distinctions between potential solutions. This practical choice of constraint interval needs to be adjusted considering the problem size, spatial units, and overall population distribution of various applications.

For further search for a complete set of Pareto-optimal solutions, the Non-Inferior Set Estimation (NISE) method can be utilized. In Figure 3, the Boundary of Unsupported Solution Search (BUSS) region is illustrated with non-dominated solutions classified into supported and unsupported categories (Medrano and Church, 2014 [36]). It confines the range of constraint bound to further explore to identify a more complete set of Pareto-optimal solutions. In this paper, there were two non-dominated solutions that dominate the current districting. An additional 10-interval constraint method was performed between the two non-dominated solutions’ maximum district population. Even though no additional non-dominated solutions were identified in this application case, this approach can be helpful with larger problem set, considering more spatial units and more districts.

Districting is vital for various aspects of human life, influencing education, business services, and political representation. Key principles for fair districting include equinumerosity, ensuring each district has an equal population; contiguity, ensuring all parts of a district are connected; and compactness, avoiding irregular shapes that serve partisan interests. The challenge lies in balancing these principles, as optimizing one often affects the others. While single-objective optimization has been common, multi-objective approaches provide Pareto-optimal solutions, offering decision makers various options. This study aims to bridge this gap by proposing a bi-objective spatial optimization model for districting that addresses these challenges comprehensively.