Measuring the Compactness of Political Districting Plans

The United States Supreme Court has long recognized compactness as an important principle in assessing the constitutionality of political districting plans. We propose a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable -- which we coin the relative proximity index. We prove that any compactness measure which satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, requiring us to solve for each state's maximal compactness; an NP-hard problem. Using two properties of maximally compact districts, we prove they are power diagrams and develop an algorithm based on these insights. The correlation between our index and the commonly-used measures of dispersion and perimeter is -.22 and -.06, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when their average vote increases is significantly greater under maximally compact districting plans, relative to the existing plans.

Police districting problem: literature review and annotated bibliography

The police districting problem concerns the efficient and effective design of patrol sectors in terms of performance attributes. Effectiveness is particularly important as it directly influences the ability of police agencies to stop and prevent crime. However, in this problem, a homogeneous distribution of workload is also desirable to guarantee fairness to the police agents and an increase in their satisfaction. This chapter provides a systematic review of the literature related to the police districting problem, whose history dates back to almost 50 years ago. Contributions are categorized in terms of attributes and solution methodology adopted. Also, an annotated bibliography that presents the most relevant elements of each research is given

A Comparison of Local Search Methods for the Multicriteria Police Districting Problem on Graph

In the current economic climate, law enforcement agencies are facing resource shortages. The effective and efficient use of scarce resources is therefore of the utmost importance to provide a high standard public safety service. Optimization models specifically tailored to the necessity of police agencies can help to ameliorate their use. The Multicriteria Police Districting Problem (MC-PDP) on a graph concerns the definition of sound patrolling sectors in a police district. The objective of this problem is to partition a graph into convex and continuous subsets, while ensuring efficiency and workload balance among the subsets. The model was originally formulated in collaboration with the Spanish National Police Corps. We propose for its solution three local search algorithms: a Simple Hill Climbing, a Steepest Descent Hill Climbing, and a Tabu Search. To improve their diversification capabilities, all the algorithms implement a multistart procedure, initialized by randomized greedy solutions. The algorithms are empirically tested on a case study on the Central District of Madrid. Our experiments show that the solutions identified by the novel Tabu Search outperform the other algorithms. Finally, research guidelines for future developments on the MC-PDP are given

Measuring the Compactness of Political Districting Plans

We develop a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable, which we coin the relative proximity index. Any compactness measure that satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, which requires us to solve for each state’s maximal compactness—a problem that is nondeterministic polynomial-time hard (NP hard). The correlations between our index and the commonly used measures of dispersion and perimeter are −.37 and −.29, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when its average vote increases is significantly greater under maximally compact districting plans relative to the existing plans.Economic

A Comparison of Local Search Methods for the Multicriteria Police Districting Problem on Graph

In the current economic climate, law enforcement agencies are facing resource shortages. The effective and efficient use of scarce resources is therefore of the utmost importance to provide a high standard public safety service. Optimization models specifically tailored to the necessity of police agencies can help to ameliorate their use. The Multicriteria Police Districting Problem (MC-PDP) on a graph concerns the definition of sound patrolling sectors in a police district. The objective of this problem is to partition a graph into convex and continuous subsets, while ensuring efficiency and workload balance among the subsets. The model was originally formulated in collaboration with the Spanish National Police Corps. We propose for its solution three local search algorithms: a Simple Hill Climbing, a Steepest Descent Hill Climbing, and a Tabu Search. To improve their diversification capabilities, all the algorithms implement a multistart procedure, initialized by randomized greedy solutions. The algorithms are empirically tested on a case study on the Central District of Madrid. Our experiments show that the solutions identified by the novel Tabu Search outperform the other algorithms. Finally, research guidelines for future developments on the MC-PDP are given

Emergency Medical Service Ambulance System Planning: History and Models

Integer linear programming models that incorporate probabilistic and stochastic components represent one approach for capturing the stochastic nature of emergency medical service ambulance systems. This includes modeling non-deterministic call arrival and servicing rates and congestion in the ambulance network (i.e., ambulance unavailability). These models focus on maximizing the total population that can find an available ambulance within a set service time standard (s) with a probability of at least α%. In MALP the concept of local vehicle busyness estimates is introduced to estimate the availability of service in a neighborhood given the neighborhood’s level of demand and the number of ambulance vehicles located in the neighborhood. QMALP is an extension of MALP where queue-theory derived parameters are implemented in the MALP model framework in order to relax the assumption that the probability of different ambulances being busy are independent. Despite this considerable development, several concerns remained about MALP and QMALP, namely the districting assumption where its assumed that a neighborhood’s calls for service are served only by an ambulance in the area, that ambulances in a neighborhood only serve calls for service originating within the neighborhood, or that at least the flow of ambulance service to and from external neighborhoods was roughly equal. Questions have been raised about the validity of MALP and QMALP’s reliability estimates, that is, whether a neighborhood actually received α-reliable service.To address these issues, we developed the Resource-Constrained Queue-based Maximum Availability Location Problem (RC-QMALP). This model is based on a location-allocation framework that (1) assigns workload from neighborhoods to ambulances located within s and ambulance idle capacity to neighborhoods and (2) includes additional constraints designed to help ensure the validity of the original MALP and QMALP constraints used to establish whether a neighborhood can find an available ambulance with α-reliability. We also implemented a secondary minsum objective that minimizes the average travel distance between ambulances and the neighborhoods they service while maintaining the priority of the MALP and QMALP coverage objective.In this thesis, we validated RC-QMALP by comparing the reliable coverage levels predicted by the RC-QMALP to the ambulance system simulations that used the locational configurations suggested by the RC-QMALP. We found that MALP 2 and QMALP provided higher levels of reliable coverage and that RC-QMALP’s secondary objective has a negligible impact on system performance. However, RC-QMALP-based models provide more accurate estimates of reliable coverage and location solutions whose simulated reliable coverage performance was always within 5% of the optimal solution with the same system parameters (we tested 1,080 different model configurations). Our work suggests that (1) simulation models must be developed to handle the modeling assumptions that underlie location optimization models and that (2) service reliability location models should consider additional factors such as ambulance workloads (and their distribution)

Equity in the Police Districting Problem: balancing territorial and racial fairness in patrolling operations

Objectives The Police Districting Problem concerns the definition of patrol districts that distribute police resources in a territory in such a way that high-risk areas receive more patrolling time than low-risk areas, according to a principle of territorial fairness. This results in patrolling configurations that are efficient and effective at controlling crime but that, at the same time, might exacerbate racial disparity in police stops and arrests. In this paper, an Equitable Police Districting Problem that combines crime-reduction effectiveness with racial fairness is proposed. The capability of this model in designing patrolling configurations that find a balance between territorial and racial fairness is assessed. Also, the trade-off between these two criteria is analyzed. Methods The Equitable Police Districting Problem is defined as a mixed-integer program. The objective function is formulated using Compromise Programming and Goal Programming. The model is validated on a real-world case study on the Central District of Madrid, Spain, and its solutions are compared to standard patrolling configurations currently used by the police. Results A trade-off between racial fairness and crime control is detected. However, the experiments show that including the proposed racial criterion in the optimization of patrol districts greatly improves racial fairness with limited detriment to the policing effectiveness. Also, the model produces solutions that dominate the patrolling configurations currently in use by the police. Conclusions The results show that the model successfully provides a quantitative evaluation of the trade-off between the criteria and is capable of defining patrolling configurations that are efficient in terms of both racial and territorial fairness

Imposing connectivity in network design problems

Imposing connectivity arises in multiple real-world problems -- telecommunication network design, social network analysis, reserve network design, and redistricting to name a few. In network optimization, connectivity constraints are usually imposed in three spaces: (i) vertex space, (ii) edge space, and (iii) vertex-and-edge space. In this dissertation, we focus on imposing connectivity in the vertex and edge spaces. We study connectivity constraints in telecommunication and redistricting networks (both in the vertex space), revisit the spanning tree polytope in planar graphs (in the edge space), and conduct a polyhedral study of k connected components (in the vertex space)

Geometric partitioning algorithms for fair division of geographic resources

University of Minnesota Ph.D. dissertation. July 2014. Major: Industrial and Systems Engineering. Advisor: John Gunnar Carlsson. 1 computer file (PDF): vi, 140 pages, appendices p. 129-140.This dissertation focuses on a fundamental but under-researched problem: how does one divide a piece of territory into smaller pieces in an efficient way? In particular, we are interested in \emph{map segmentation problem} of partitioning a geographic region into smaller subregions for allocating resources or distributing a workload among multiple agents. This work would result in useful solutions for a variety of fundamental problems, ranging from congressional districting, facility location, and supply chain management to air traffic control and vehicle routing. In a typical map segmentation problem, we are given a geographic region $R$, a probability density function defined on $R$ (representing, say population density, distribution of a natural resource, or locations of clients) and a set of points in $R$ (representing, say service facilities or vehicle depots). We seek a \emph{partition} of $R$ that is a collection of disjoint sub-regions $\{R_1, . . . , R_n\}$ such that $\bigcup_i R_i = R$, that optimizes some objective function while satisfying a shape condition. As examples of shape conditions, we may require that all sub-regions be compact, convex, star convex, simply connected (not having holes), connected, or merely measurable.Such problems are difficult because the search space is infinite-dimensional (since we are designing boundaries between sub-regions) and because the shape conditions are generally difficult to enforce using standard optimization methods. There are also many interesting variants and extensions to this problem. It is often the case that the optimal partition for a problem changes over time as new information about the region is collected. In that case, we have an \emph{online} problem and we must re-draw the sub-region boundaries as time progresses. In addition, we often prefer to construct these sub-regions in a \emph{decentralized} fashion: that is, the sub-region assigned to agent $i$ should be computable using only local information to agent $i$ (such as nearby neighbors or information about its surroundings), and the optimal boundary between two sub-regions should be computable using only knowledge available to those two agents.This dissertation is an attempt to design geometric algorithms aiming to solve the above mentioned problems keeping in view the various design constraints. We describe the drawbacks of the current approach to solving map segmentation problems, its ineffectiveness to impose geometric shape conditions and its limited utility in solving the online version of the problem. Using an intrinsically interdisciplinary approach, combining elements from variational calculus, computational geometry, geometric probability theory, and vector space optimization, we present an approach where we formulate the problems geometrically and then use a fast geometric algorithm to solve them. We demonstrate our success by solving problems having a particular choice of objective function and enforcing certain shape conditions. In fact, it turns out that such methods actually give useful insights and algorithms into classical location problems such as the continuous $k$-medians problem, where the aim is to find optimal locations for facilities. We use a map segmentation technique to present a constant factor approximation algorithm to solve the continuous $k$-medians problem in a convex polygon. We conclude this thesis by describing how we intend to build on this success and develop algorithms to solve larger classes of these problems