Redividing the Cake

A heterogeneous resource, such as a land-estate, is already divided among several agents in an unfair way. It should be re-divided among the agents in a way that balances fairness with ownership rights. We present re-division protocols that attain various trade-off points between fairness and ownership rights, in various settings differing in the geometric constraints on the allotments: (a) no geometric constraints; (b) connectivity --- the cake is a one-dimensional interval and each piece must be a contiguous interval; (c) rectangularity --- the cake is a two-dimensional rectangle or rectilinear polygon and the pieces should be rectangles; (d) convexity --- the cake is a two-dimensional convex polygon and the pieces should be convex. Our re-division protocols have implications on another problem: the price-of-fairness --- the loss of social welfare caused by fairness requirements. Each protocol implies an upper bound on the price-of-fairness with the respective geometric constraints.Comment: Extended IJCAI 2018 version. Previous name: "How to Re-Divide a Cake Fairly

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

Constraint-aware coordinated construction of generic structures

This paper presents a constraint-aware decentralized approach to construction with teams of robots. We present an extension to existing work on a distributed controller for robotic construction of simple structures. Our previous work described a set of adaptive algorithms for constructing truss structures given a target geometry using continuous and graph-based equal-mass partitioning [1], [2]. Using this work as a foundation, we present an algorithm which performs construction tasks and conforms to physical constraints while considering those constraints to parallelize tasks. This is accomplished by defining a mass function which reflects the priority of part placement and prevents physically impossible states. This mass function generates a set of pointmasses in ℝn, and we present a novel algorithm for finding a locally optimal, equal-mass, convex tessellation of such a set.Boeing CompanyNational Science Foundation (U.S.).National Science Foundation (U.S.). Office of Emerging Frontiers in Research and Innovation (Grant #0735953)United States. Army Research Office. Multidisciplinary University Research Initiative. Swarms of Autonomous Robots and Mobile Sensors Project (Grant number N0014-09-1051)United States. Army Research Office. Multidisciplinary University Research Initiative. Scalable (Grant number 544252

Partitioning a Polygon Into Small Pieces

We study the problem of partitioning a given simple polygon $P$ into a minimum number of polygonal pieces, each of which has bounded size. We give algorithms for seven notions of `bounded size,' namely that each piece has bounded area, perimeter, straight-line diameter, geodesic diameter, or that each piece must be contained in a unit disk, an axis-aligned unit square or an arbitrarily rotated unit square. A more general version of the area problem has already been studied. Here we are, in addition to $P$, given positive real values $a_1,\ldots,a_k$ such that the sum $\sum_{i=1}^k a_i$ equals the area of $P$. The goal is to partition $P$ into exactly $k$ pieces $Q_1,\ldots,Q_k$ such that the area of $Q_i$ is $a_i$. Such a partition always exists, and an algorithm with running time $O(nk)$ has previously been described, where $n$ is the number of corners of $P$. We give an algorithm with optimal running time $O(n+k)$. For polygons with holes, we get running time $O(n\log n+k)$. For the other problems, it seems out of reach to compute optimal partitions for simple polygons; for most of them, even in extremely restricted cases such as when $P$ is a square. We therefore develop $O(1)$-approximation algorithms for these problems, which means that the number of pieces in the produced partition is at most a constant factor larger than the cardinality of a minimum partition. Existing algorithms do not allow Steiner points, which means that all corners of the produced pieces must also be corners of $P$. This has the disappointing consequence that a partition does often not exist, whereas our algorithms always produce useful partitions. Furthermore, an optimal partition without Steiner points may require $\Omega(n)$ pieces for polygons where a partition consisting of just $2$ pieces exists when Steiner points are allowed.Comment: 32 pages, 24 figure

Multi-Robot Persistent Coverage in Complex Environments

Los recientes avances en robótica móvil y un creciente desarrollo de robots móviles asequibles han impulsado numerosas investigaciones en sistemas multi-robot. La complejidad de estos sistemas reside en el diseño de estrategias de comunicación, coordinación y controlpara llevar a cabo tareas complejas que un único robot no puede realizar. Una tarea particularmente interesante es la cobertura persistente, que pretende mantener cubierto en el tiempo un entorno con un equipo de robots moviles. Este problema tiene muchas aplicaciones como aspiración o limpieza de lugares en los que la suciedad se acumula constantemente, corte de césped o monitorización ambiental. Además, la aparición de vehículos aéreos no tripulados amplía estas aplicaciones con otras como la vigilancia o el rescate.Esta tesis se centra en el problema de cubrir persistentemente entornos progresivamente mas complejos. En primer lugar, proponemos una solución óptima para un entorno convexo con un sistema centralizado, utilizando programación dinámica en un horizonte temporalnito. Posteriormente nos centramos en soluciones distribuidas, que son más robustas, escalables y eficientes. Para solventar la falta de información global, presentamos un algoritmo de estimación distribuido con comunicaciones reducidas. Éste permite a los robots teneruna estimación precisa de la cobertura incluso cuando no intercambian información con todos los miembros del equipo. Usando esta estimación, proponemos dos soluciones diferentes basadas en objetivos de cobertura, que son los puntos del entorno en los que más se puedemejorar dicha cobertura. El primer método es un controlador del movimiento que combina un término de gradiente con un término que dirige a los robots hacia sus objetivos. Este método funciona bien en entornos convexos. Para entornos con algunos obstáculos, el segundométodo planifica trayectorias abiertas hasta los objetivos, que son óptimas en términos de cobertura. Finalmente, para entornos complejos no convexos, presentamos un algoritmo capaz de encontrar particiones equitativas para los robots. En dichas regiones, cada robotplanifica trayectorias de longitud finita a través de un grafo de caminos de tipo barrido.La parte final de la tesis se centra en entornos discretos, en los que únicamente un conjunto finito de puntos debe que ser cubierto. Proponemos una estrategia que reduce la complejidad del problema separándolo en tres subproblemas: planificación de trayectoriascerradas, cálculo de tiempos y acciones de cobertura y generación de un plan de equipo sin colisiones. Estos subproblemas más pequeños se resuelven de manera óptima. Esta solución se utiliza en último lugar para una novedosa aplicación como es el calentamiento por inducción doméstico con inductores móviles. En concreto, la adaptamos a las particularidades de una cocina de inducción y mostramos su buen funcionamiento en un prototipo real.Recent advances in mobile robotics and an increasing development of aordable autonomous mobile robots have motivated an extensive research in multi-robot systems. The complexity of these systems resides in the design of communication, coordination and control strategies to perform complex tasks that a single robot can not. A particularly interesting task is that of persistent coverage, that aims to maintain covered over time a given environment with a team of robotic agents. This problem is of interest in many applications such as vacuuming, cleaning a place where dust is continuously settling, lawn mowing or environmental monitoring. More recently, the apparition of useful unmanned aerial vehicles (UAVs) has encouraged the application of the coverage problem to surveillance and monitoring. This thesis focuses on the problem of persistently covering a continuous environment in increasingly more dicult settings. At rst, we propose a receding-horizon optimal solution for a centralized system in a convex environment using dynamic programming. Then we look for distributed solutions, which are more robust, scalable and ecient. To deal with the lack of global information, we present a communication-eective distributed estimation algorithm that allows the robots to have an accurate estimate of the coverage of the environment even when they can not exchange information with all the members of the team. Using this estimation, we propose two dierent solutions based on coverage goals, which are the points of the environment in which the coverage can be improved the most. The rst method is a motion controller, that combines a gradient term with a term that drives the robots to the goals, and which performs well in convex environments. For environments with some obstacles, the second method plans open paths to the goals that are optimal in terms of coverage. Finally, for complex, non-convex environments we propose a distributed algorithm to nd equitable partitions for the robots, i.e., with an amount of work proportional to their capabilities. To cover this region, each robot plans optimal, nite-horizon paths through a graph of sweep-like paths. The nal part of the thesis is devoted to discrete environment, in which only a nite set of points has to be covered. We propose a divide-and-conquer strategy to separate the problem to reduce its complexity into three smaller subproblem, which can be optimally solved. We rst plan closed paths through the points, then calculate the optimal coverage times and actions to periodically satisfy the coverage required by the points, and nally join together the individual plans of the robots into a collision-free team plan that minimizes simultaneous motions. This solution is eventually used for a novel application that is domestic induction heating with mobile inductors. We adapt it to the particular setting of a domestic hob and demonstrate that it performs really well in a real prototype.<br /

A Decomposition Strategy for Optimal Coverage of an Area of Interest using Heterogeneous Team of UAVs

In this thesis, we study the problem of optimal search and coverage with heterogeneous team of unmanned aerial vehicles (UAVs). The team must complete the coverage of a given region while minimizing the required time and fuel for performing the mission. Since the UAVs have different characteristics one needs to equalize the ratio of the task to the capabilities of each agent to obtain an optimal solution. A multi-objective task assignment framework is developed for finding the best group of agents that by assigning the optimal tasks would carry out the mission with minimum total cost. Once the optimal tasks for UAVs are obtained, the coverage area (map) is partitioned according to the results of the multi-objective task assignment. The strategy is to partition the coverage area into separate regions so that each agent is responsible for performing the surveillance of its particular region. The decentralized power diagram algorithm is used to partition the map according to the results of the task assignment phase. Furthermore, a framework for solving the task assignment problem and the coverage area partitioning problem in parallel is proposed. A criterion for achieving the minimum number of turns in covering a region a with single UAV is studied for choosing the proper path direction for each UAV. This criterion is extended to develop a method for partitioning the coverage area among multiple UAVs that minimizes the number of turns. We determine the "best" team for performing the coverage mission and we find the optimal workload for each agent that is involved in the mission through a multi-objective optimization procedure. The search area is then partitioned into disjoint subregions, and each agent is assigned to a region having an optimum area resulting in the minimum cost for the entire surveillance mission