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

Approximating Median Points in a Convex Polygon

We develop two simple and efficient approximation algorithms for the continuous $k$-medians problems, where we seek to find the optimal location of $k$ facilities among a continuum of client points in a convex polygon $C$ with $n$ vertices in a way that the total (average) Euclidean distance between clients and their nearest facility is minimized. Both algorithms run in $\mathcal{O}(n + k + k \log n)$ time. Our algorithms produce solutions within a factor of 2.002 of optimality. In addition, our simulation results applied to the convex hulls of the State of Massachusetts and the Town of Brookline, MA show that our algorithms generally perform within a range of 5\% to 22\% of optimality in practice

An efficient algorithm for commercial aircraft trajectory optimization in the air traffic system.

University of Minnesota M.S. thesis. July 2012. Major: Aerospace Engineering and Mechanics. Advisor: Yiyuan J. Zhao. 1 computer file (PDF); vii, 64 pages.A discrete search strategy is presented to determine optimal aircraft trajectories which can be unconstrained or regulated to follow current Air Traffic Control (ATC) procedures. The heuristic based Astar (A*) search algorithm has been selected for its efficiency and its inherent ability to handle numerous constraints as a discrete method. A point-mass aircraft model is assumed to accurately simulate commercial aircraft dynamics for the provided trajectories. The two dimensional space and the states of aircraft have been divided into discrete pieces. To show the effectiveness of the algorithm, two-dimensional vertical and horizontal profile are simulated. Simulation results compare optimal trajectories that range from unconstrained to those that completely adhere to strict ATC procedures. Those trajectories following ATC procedures follow a segmented flight pattern where each segment follows specified objectives, terminating when certain criteria has been met. Trajectories are optimized for a combination of time and fuel with an emphasis on reducing fuel consumption