Approximation Schemes for Maximum Weight Independent Set of Rectangles

In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1+epsilon)-approximation algorithm for the MWISR problem with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n / log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to Arora et al. this allows us to upper bound our approximation ratio by (1+epsilon). Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1+epsilon)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem

On the Parameterized Complexity of Red-Blue Points Separation

We study the following geometric separation problem: Given a set R of red points and a set B of blue points in the plane, find a minimum-size set of lines that separate R from B. We show that, in its full generality, parameterized by the number of lines k in the solution, the problem is unlikely to be solvable significantly faster than the bruteforce nO(k) -time algorithm, where n is the total number of points. Indeed, we show that an algorithm running in time f(k)nᵒ(k/log k) , for any computable function f, would disprove ETH. Our reduction crucially relies on selecting lines from a set with a large number of different slopes (i.e., this number is not a function of k). Conjecturing that the problem variant where the lines are required to be axis-parallel is FPT in the number of lines, we show the following preliminary result. Separating R from B with a minimum-size set of axis-parallel lines is FPT in the size of either set, and can be solved in time O∗(9|B|) (assuming that B is the smaller set)

Studies of optimal track-fitting techniques for the DarkLight experiment

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (page 49).The DarkLight experiment is searching for a dark force carrier, the A' boson, and hopes to measure its mass with a resolution of approximately 1 MeV/c 2 . This mass calculation requires precise reconstruction to turn data, in the form of hits within the detector, into a particle track with known initial momentum. This thesis investigates the appropriateness of the Billoir optimal fit to reconstruct helical, low-energy lepton tracks while accounting for multiple scattering, using two separate track parameterizations. The first method approximates the track as a piecewise concatenation of parabolas in three-dimensions, and (wrongly) assumes that the y and z components of the track are independent. When tested using simulated data, this returns a track which geometrically fits the data. However, the momentum extracted from this geometrical representation is an order of magnitude higher than the true momentum of the track. The second method approximates the track as a piecewise concatenation of helical segments. This returns a track which geometrically fits the data even better than the parabolic parameterization, but which returns a momentum which depends on the seeds to the algorithm. Some further work must be done to modify this fitting method so that it will reliably reconstruct tracks.by Purnima Parvathy Balakrishnan.S.B

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

Proceedings of the 84th European Study Group Mathematics with Industry (SWI 2012), Eindhoven, January 30 - February 3, 2012

Introduction There are a few welldefined moments when mathematicians can get in contact with relevant unsolved problems proposed by the industry. One such a moment is the socalled "Study Group". The concept of the Study Group is rather simple and quite efficient: A group of mathematicians (of very different expertise) work together for one week. As a rule, on a Monday the industrial problems are presented by their owners, then few research groups selforganize around the proposed problems and work intensively until Friday, when the main findings are presented. The insight obtained via mathematical modeling together with the transfer of suitable mathematical technology usually lead the groups to adequate approximate solutions. As a direct consequence of this fact, the problem owners often decide to benefit more from such knowledge transfer and suggest related followup projects. In the period January 31– February 3, 2012, it was the turn of the Department of Mathematics and Computer Science of the Eindhoven University of Technology to organize and to host the "Studiegroep Wiskunde met de Industrie/Study Group Mathematics with the Industry" (shortly: SWI 2012, but also referred to as ESG 84, or as the 84th European Study Group with Industry). This was the occasion when about 80 mathematicians enjoyed working on six problems. Most of the participants were coming from a Dutch university, while a few were from abroad (e.g. from UK, Germany, France, India, Russia, Georgia, Turkey, India, and Sri Lanka). The open industrial problems were proposed by Endinet, Philips Lighting, Thales, Marin, Tata Steel, and Bartels Engineering. Their solutions are shown in this proceedings. They combine ingenious mathematical modeling with specific mathematical tools like geometric algorithms, combinatorial optimization of networks, identification of parameters and model structures, probability theory, and statistical data analysis. It is worth mentioning that this scientific proceedings is accompanied by a popular proceedings, written by Ionica Smeets, containing layman’s descriptions of the proposed problems and of the corresponding results

Proceedings of the 84th European Study Group Mathematics with Industry (SWI 2012), Eindhoven, January 30 - February 3, 2012

Introduction There are a few welldefined moments when mathematicians can get in contact with relevant unsolved problems proposed by the industry. One such a moment is the socalled "Study Group". The concept of the Study Group is rather simple and quite efficient: A group of mathematicians (of very different expertise) work together for one week. As a rule, on a Monday the industrial problems are presented by their owners, then few research groups selforganize around the proposed problems and work intensively until Friday, when the main findings are presented. The insight obtained via mathematical modeling together with the transfer of suitable mathematical technology usually lead the groups to adequate approximate solutions. As a direct consequence of this fact, the problem owners often decide to benefit more from such knowledge transfer and suggest related followup projects. In the period January 31– February 3, 2012, it was the turn of the Department of Mathematics and Computer Science of the Eindhoven University of Technology to organize and to host the "Studiegroep Wiskunde met de Industrie/Study Group Mathematics with the Industry" (shortly: SWI 2012, but also referred to as ESG 84, or as the 84th European Study Group with Industry). This was the occasion when about 80 mathematicians enjoyed working on six problems. Most of the participants were coming from a Dutch university, while a few were from abroad (e.g. from UK, Germany, France, India, Russia, Georgia, Turkey, India, and Sri Lanka). The open industrial problems were proposed by Endinet, Philips Lighting, Thales, Marin, Tata Steel, and Bartels Engineering. Their solutions are shown in this proceedings. They combine ingenious mathematical modeling with specific mathematical tools like geometric algorithms, combinatorial optimization of networks, identification of parameters and model structures, probability theory, and statistical data analysis. It is worth mentioning that this scientific proceedings is accompanied by a popular proceedings, written by Ionica Smeets, containing layman’s descriptions of the proposed problems and of the corresponding results