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

Certified Approximation Algorithms for the Fermat Point and n-Ellipses

Given a set A of n points in ?^d with weight function w: A??_{> 0}, the Fermat distance function is ?(x): = ?_{a?A}w(a)?x-a?. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function ?. We consider the problem of computing a point x?* that is an ?-approximation of x* in the sense that ?x?*-x*? ?(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques

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

On the Strategyproofness of the Geometric Median

The geometric median of a tuple of vectors is the vector that minimizes the sum of Euclidean distances to the vectors of the tuple. Classically called the Fermat-Weber problem and applied to facility location, it has become a major component of the robust learning toolbox. It is typically used to aggregate the (processed) inputs of different data providers, whose motivations may diverge, especially in applications like content moderation. Interestingly, as a voting system, the geometric median has well-known desirable properties: it is a provably good average approximation, it is robust to a minority of malicious voters, and it satisfies the "one voter, one unit force" fairness principle. However, what was not known is the extent to which the geometric median is strategyproof. Namely, can a strategic voter significantly gain by misreporting their preferred vector? We prove in this paper that, perhaps surprisingly, the geometric median is not even $\alpha$-strategyproof, where $\alpha$ bounds what a voter can gain by deviating from truthfulness. But we also prove that, in the limit of a large number of voters with i.i.d. preferred vectors, the geometric median is asymptotically $\alpha$-strategyproof. We show how to compute this bound $\alpha$. We then generalize our results to voters who care more about some dimensions. Roughly, we show that, if some dimensions are more polarized and regarded as more important, then the geometric median becomes less strategyproof. Interestingly, we also show how the skewed geometric medians can improve strategyproofness. Nevertheless, if voters care differently about different dimensions, we prove that no skewed geometric median can achieve strategyproofness for all. Overall, our results constitute a coherent set of insights into the extent to which the geometric median is suitable to aggregate high-dimensional disagreements.Comment: 55 pages, 7 figure

Konvexgeometrie

[no abstract available

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend