DOCSLIB.ORG

The NP-Hardness of Covering Points with Lines, Paths and Tours and Their Tractability with FPT-Algorithms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The NP-Hardness of Covering Points with Lines, Paths and Tours and Their Tractability with FPT-Algorithms The NP-Hardness of Covering Points with Lines, Paths and Tours and their Tractability with FPT-Algorithms by Apichat Heednacram B. Eng. (1st Class Hons), Griffith University, 2001 Submitted in fulfilment of the requirements of the degree of Doctor of Philosophy Institute for Integrated and Intelligent Systems Science, Environment, Engineering and Technology Griffith University March 2010 i Abstract Given a problem for which no polynomial-time algorithm is likely to exist, we investigate how to attack this seemingly intractable problem based on parame- terized complexity theory. We study hard geometric problems, and show that they are fixed-parameter tractable (FPT) given an instance and a parameter k. This allows the problems to be solved exactly, rather than approximately, in polynomial time in the size of the input and exponential time in the parameter. Although the parameterized approach is still young, in recent years, there have been many results published concerning graph problems and databases. However, not many earlier results apply the parameterized approach in the field of computational geometry. This thesis, therefore, focuses on geometric NP-hard problems. These problems are the Line Cover problem, the Rectilinear Line Cover problem in higher dimensions, the Rectilinear Minimum-Links Spanning Path problem in higher dimensions, the Rectilinear Hyper- plane Cover problem, the Minimum-Bends Traveling Salesman Prob- lem and the Rectilinear Minimum-Bends Traveling Salesman Prob- lem, in addition to some other variants of these problems. The Rectilinear Minimum-Links Spanning Path problem in higher dimensions and the Rectilinear Hyperplane Cover problem had been the subject of only conjectures about their intractability. Therefore, we present the NP-completeness proofs for these problems. After verifying their hardness, we use the fixed-parameter approach to solve the two problems. We focus on solving the decision version of the problems, rather than solving the optimizations. However, with the Line Cover problem we demonstrate that it is not difficult to adapt algorithms for the decision version to algorithms for the optimization version. We also implement several algorithms for the Line Cover problem and conduct experimental evaluations of our algorithms with respect to previously known algorithms. For the remaining problems in the thesis, we will establish only the fundamental results. That is, we determine fixed-parameter tractability of those problems. ii iii Statement of Originality This work has not previously been submitted for a degree or diploma in any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made in the thesis itself. (Signed) Apichat Heednacram iv v c Copyright 2010 Apichat Heednacram vi vii Approval Name: Apichat Heednacram Student No: 964128 Degree: Doctor of Philosophy Thesis Title: The NP-Hardness of Covering Points with Lines, Paths and Tours and their Tractability with FPT- Algorithms Submission Date: 01 March 2010 Principal Supervisor: Professor Vladimir Estivill-Castro School of Information and Communication Technology, Griffith University Australia Principal Supervisor: Dr. Francis Suraweera School of Information and Communication Technology, Griffith University Australia viii ix Acknowledgements I would like to thank the Australian Government, Department of Education, Em- ployment and Workplace Relations (DEEWR) for awarding me the Endeavour Postgraduate Award. The scholarship was generously supplied for three years, thus enabling me to enjoy my research with sufficient financial support. I express my sincere gratitude to my two supervisors, Dr Francis Suraweera and Professor Vladimir Estivill-Castro. Without their support, this thesis would not have been possible. I cannot thank them enough for their guidance over the years, in particular, their commitment and time in making weekly discussions extremely useful. I would also like to acknowledge Professor Vladimir Estivill-Castro for his financial support for my travels overseas and for the many opportunities he has provided. Thanks for providing the sponsorship to conferences such as ISAAC08, CATS08, CATS09 and particularly, a visit to the Yahoo Research and the Barcelona Media Innovation Centre (FBM-UPF) at the Universitat Pompeu Fabra (UPF) in Barcelona, Spain. Thank you Professor Josep Blat, Director of Departament de Tecnologia for providing a working space for several weeks at the UPF. I also thank Professor Abdul Sattar, Director of the Institute for Integrated and Intelligent Systems (IIIS) for helping me on several occa- sions with the VISA application and travel-documents. I would like to thank the Enterprise Information Infrastructure (EII) for providing sponsorship to the Theoretical Computer Science Day in Sydney and the EII PhD School. Thanks also to Kathleen Williamson, The University of Queensland, for making each trip as convenient as possible. Thanks to Mike Fellows and Frances Rosamond for the few meetings we had and thank you for keeping me updated about the FPT news. I would like to thank Rod Downey for providing some interesting papers during the ACSW2009 in New Zealand. I thank Joel Fenwick and Artak Amirbekyan for helping me with the initial stages of the PhD. Thank you Mahdi Parsa, my fellow PhD student, who shares the same passion about parameterized complexity theory. Finally, thanks to my family and friends for their unconditional love. x xi To Lord Buddha who told us that our innate wisdom and virtuous abilities are not truly lost, just not yet uncovered. xii xiii List of Outcomes Arising from this Thesis International journals with papers fully refereed 1. V. Estivill-Castro, A. Heednacram, and F. Suraweera. Reduction Rules Deliver Efficient FPT-Algorithms for Covering Points with Lines. ACM Journal of Experimental Algorithmics, 14:1.7–1.26, November 2009. 2. V. Estivill-Castro, A. Heednacram, and F. Suraweera. NP-completeness and FPT Results for Rectilinear Covering Problems. Journal of Universal Computer Science, 16(5):622–652, May 2010. 3. V. Estivill-Castro, A. Heednacram, and F. Suraweera. FPT-algorithms for Minimum-Bends Tours. International Journal of Computational Geometry and Applications, August 2010 (Accepted with minor corrections). International conferences with papers fully refereed 1. V. Estivill-Castro, A. Heednacram, and F. Suraweera. The Rectilinear k- Bends TSP. In, M. T. Thai and S. Sahni, editors, Proceedings of the 16th Annual International Combinatorics Conference (COCOON), volume 6196 of Lecture Notes in Computer Science, pages 264–277. Springer, Berlin, July 19–21, 2010. xiv xv Notation S — a set of input points. n — the size of S. S — a subset of S; that is, S S. ⊆ n — the size of S where n n. ≤ P — the class of polynomial-time solvable problems. Q — a parameterized problem. k — a positive integer and a parameter of parameterized problems. d — the number of dimensions. φ — the number of orientations. L3 — the set of all lines that cover at least 3 points in S. L n/k — the set of all lines that cover at least n/k points in S. Lk+1 — the set of all lines that cover at least k + 1 points in S. L — the cardinality of a set L. | | cover(L) — all points in S covered by some line in L. lines(T ) — the set of lines used by the line-segments in a tour T . p1,p2 — the line through points p1 and p2. p1p2 — the line-segment between points p1 and p2. d(p1,p2) — the distance from point p1 to p2. (L) — the arrangement of lines induced by L. A dual(p) — a line in dual space to the point p in primal space. Rd — the real coordinate space of d-dimensions. Π(x, , ) — the yz-plane. ∗ ∗ Π( ,y, ) — the xz-plane. ∗ ∗ Π( , ,z) — the xy-plane. ∗ ∗ xvi xvii Glossary of Terms Confidence Interval: A confidence interval indicates that the expected run- ning time of the algorithm on an instance has 95% probability of falling inside the interval. A confidence interval is defined by the following for- mula: 95%C.I. = M (1.96 SD/√N) where M is the sample mean, SD is the standard deviation,± and∗ N is the number of samples. If the confi- dence intervals are disjoint, then we have statistical significance that one algorithm’s CPU time is lesser than the other. Direction: We use this term for the direction of travel of a line-segment in a tour; for example, a vertical line-segment can be traveled North-South or in the opposite direction, South-North. Dual Space: The dual-space mapping assigns a line l given by y = mx + b in a 2-dimensional space to the point pl =(m, b), and a point p =(px,py)to a line l given by y = p x p . This mapping− has the property that two p x − y lines lp and lq in dual space intersect at a point pl, which, in primal space, is the line l through the points p and q that are images of the two lines. Fan-out: The number of children nodes in a parent node in a tree. FPT: This denotes the class of fixed-parameter tractable parameterized prob- lems. Hard Problems: The problems for which no polynomial-time algorithms are known. In this thesis, the problems discussed are in the class of NP-hard. Hyperplane: A generalization of the concept of a plane in geometry into a larger number of dimensions. Line: An unbounded infinite set of points with zero-width that is convex and contains the shortest path between any of two points in it. Alternatively, a linear vector space of dimension 1. Line-segment: The bounded portion of a line that constitutes the shortest path between the extremes of the segment. Line Cover: A line cover is a set of lines that cover the points in S. Link Length: The number of line-segments in a given path. Orientation: The slope of a line with respect to the x-axis and, when applied to a line segment, it refers to the orientation of the line hosting it.
Load more

Recommended publications
  • Integer Programming Models for the Quasi-Polyomino Strip Packing Problem
    Irregular and quasi-polyomino strip packing problems Marcos Okamura Rodrigues Tese de Doutorado do Programa de Pós-Graduação em Ciências de Computação e Matemática Computacional (PPG-CCMC) UNIVERSIDADE DE SÃO PAULO DE SÃO UNIVERSIDADE Instituto de Ciências Matemáticas e de Computação Instituto Matemáticas de Ciências SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________ Marcos Okamura Rodrigues Irregular and quasi-polyomino strip packing problems Doctoral dissertation submitted to the Institute of Mathematics and Computer Sciences – ICMC-USP, in partial fulfillment of the requirements for the degree of the Doctorate Program in Computer Science and Computational Mathematics. FINAL VERSION Concentration Area: Computer Science and Computational Mathematics Advisor: Franklina Maria Bragion de Toledo USP – São Carlos May 2020 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados inseridos pelo(a) autor(a) Rodrigues, Marcos Okamura R685i Irregular and quasi-polyomino strip packing problems / Marcos Okamura Rodrigues; orientadora Franklina Maria Bragion de Toledo. -- São Carlos, 2020. 90 p. Tese (Doutorado - Programa de Pós-Graduação em Ciências de Computação e Matemática Computacional) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2020. 1. Strip packing. 2. Nesting. 3. Irregular shapes. 4. Quasi-polyomino. 5. Polyomino. I. Toledo, Franklina Maria Bragion de, orient. II. Título. Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176 Marcos Okamura Rodrigues Problemas de empacotamento em faixa de itens irregulares e quasi-poliominós Tese apresentada ao Instituto de Ciências Matemáticas e de Computação – ICMC-USP, como parte dos requisitos para obtenção do título de Doutor em Ciências – Ciências de Computação e Matemática Computacional.
    [Show full text]
  • Covering Orthogonal Polygons with Star Polygons: the Perfect Graph Approach
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMPUTER AND SYSTEM SCIENCES 40, 1948 (1990) Covering Orthogonal Polygons with Star Polygons: The Perfect Graph Approach RAJEEV MOTWANI* Computer Science Department, Stanford University, Stanford, California 94305 ARVIND RAGHLJNATHAN~AND HUZUR SARAN~ Computer Science Division, University of California, 573 Evans Hall, Berkeley, California 94720 Received November 29, 1988 This paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point 4 in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a mini- mum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an o(ns) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n’) algorithm. 0 1990 Academic Press, Inc. 1. INTRODUCTION One of the most well-studied class of problems in computational geometry con- cerns the notion of visibility.
    [Show full text]
  • Visibility Algorithms in the Plane
    This page intentionally left blank iii VISIBILITY ALGORITHMS IN THE PLANE A human observer can effortlessly identify visible portions of geometric objects present in the environment. However, computations of visible portions of objects from a viewpoint involving thousands of objects is a time-consuming task even for high-speed computers. To solve such visibility problems, efficient algorithms have been designed. This book presents some of these visibility algorithms in two dimensions. Specifically, basic algorithms for point visibility, weak visibility, shortest paths, visibility graphs, link paths, and visibility queries are all discussed. Several geometric properties are also established through lemmas and theorems. With over 300 figures and hundreds of exercises, this book is ideal for graduate students and researchers in the field of computational geometry. It will also be useful as a reference for researchers working in algorithms, robotics, computer graphics, and geometric graph theory, and some algorithms from the book can be used in a first course in computational geometry. Subir Kumar Ghosh is a professor of computer science at the Tata Institute of Fundamental Research, Mumbai, India and is a fellow of the Indian Academy of Sciences. He is the author of around 40 papers in the fields of computational geometry and graph theory and has worked as a visiting scientist in many reputed universities and research institutes around the world. i iii VISIBILITY ALGORITHMS IN THE PLANE SUBIR KUMAR GHOSH School of Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India iii CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521875745 © S.
    [Show full text]
  • 30 POLYGONS Joseph O’Rourke, Subhash Suri, and Csaba D
    30 POLYGONS Joseph O'Rourke, Subhash Suri, and Csaba D. T´oth INTRODUCTION Polygons are one of the fundamental building blocks in geometric modeling, and they are used to represent a wide variety of shapes and figures in computer graph- ics, vision, pattern recognition, robotics, and other computational fields. By a polygon we mean a region of the plane enclosed by a simple cycle of straight line segments; a simple cycle means that nonadjacent segments do not intersect and two adjacent segments intersect only at their common endpoint. This chapter de- scribes a collection of results on polygons with both combinatorial and algorithmic flavors. After classifying polygons in the opening section, Section 30.2 looks at sim- ple polygonizations, Section 30.3 covers polygon decomposition, and Section 30.4 polygon intersection. Sections 30.5 addresses polygon containment problems and Section 30.6 touches upon a few miscellaneous problems and results. 30.1 POLYGON CLASSIFICATION Polygons can be classified in several different ways depending on their domain of application. In chip-masking applications, for instance, the most commonly used polygons have their sides parallel to the coordinate axes. GLOSSARY Simple polygon: A closed region of the plane enclosed by a simple cycle of straight line segments. Convex polygon: The line segment joining any two points of the polygon lies within the polygon. Monotone polygon: Any line orthogonal to the direction of monotonicity inter- sects the polygon in a single connected piece. Star-shaped polygon: The entire polygon is visible from some point inside the polygon. Orthogonal polygon: A polygon with sides parallel to the (orthogonal) coordi- nate axes.
    [Show full text]
  • Cover-Decomposition and Polychromatic Numbers∗
    COVER-DECOMPOSITION AND POLYCHROMATIC NUMBERS∗ BELA´ BOLLOBAS´ † , DAVID PRITCHARD‡, THOMAS ROTHVOSS§ , AND ALEX SCOTT¶ Abstract. A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge- covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VC-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover. Key words. colouring, hypergraphs, probabilistic methods, iterated LP relaxation AMS subject classifications. 05C15, 05D05 1. Introduction. In a set system on vertex set V , a subsystem is a set cover if each vertex of V appears in at least 1 set of the subsystem. Suppose that in the whole system, each vertex appears in at least δ sets, for some large δ; does it follow that we can partition the system into 2 subsystems, such that each subsystem is a set cover? Many natural families of set systems admit a universal constant δ for which this question has an affirmative answer. Such families are called cover-decomposable. But the family of all set systems is not cover-decomposable, as the following example shows.
    [Show full text]
  • Cover-Decomposition and Polychromatic Numbers
    Cover-Decomposition and Polychromatic Numbers∗ B´ela Bollob´as† David Pritchard‡ Thomas Rothvoߧ Alex Scott¶ September 10, 2018 Abstract A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VC-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover. 1 Introduction In a set system on vertex set V , a subsystem is a set cover if each vertex of V appears in at least 1 set of the subsystem. Suppose that in the whole system, each vertex appears in at least δ sets, for some large δ; does it follow that we can partition the system into 2 subsystems, such that each subsystem is a set cover? Many natural families of set systems admit a universal constant δ for which this question has an af- firmative answer. Such families are called cover-decomposable. But the family of all set systems is not cover-decomposable, as the following example shows. For any positive integer k, consider a set system which has 2k 1 sets, and where every k sets contain one mutually common vertex not contained by the other k 1 sets.− This system satisfies the hypothesis of the question for δ = k.
    [Show full text]
  • Geometric Optimization for Classification Problems
    Geometric Optimization for Classification Problems Pablo Perez´ Lantero 2010 Universidad de Sevilla Escuela Tecnica´ Superior de Ingenieros Dpto. Matematica´ Aplicada II Geometric Optimization for Classification Problems Memoria de tesis doctoral presentada por Pablo Perez´ Lantero para optar al grado de Doctor en Matematicas´ por la Universidad de Sevilla Director: Jose´ Miguel D´ıaz-Ba´nez˜ Sevilla, Abril de 2010 Universidad de Sevilla Memoria de tesis doctoral para optar al grado de Doctor en Matematicas´ Autor: Pablo Perez´ Lantero T´ıtulo: Geometric Optimization for Classification Problems Departamento: Matematica´ Aplicada II V◦ B◦ Director: Jose´ Miguel D´ıaz-Ba´nez˜ El autor: Pablo Perez´ Lantero A mis padres, a Diana. Acknowledgements / Agradecimientos First of all, I would like to thank Jos´eMiguel D´ıaz-B´a˜nez for being my advisor. Throughout these years, he has provided me with guidance in research, valuable knowledge, and financial support. I am indebted to him. I give thanks to Inma Ventura for being an excellent partner. We spent a lot of time discussing problems, writing results, and reviewing papers. I am grateful to have collaborated with Jorge Urrutia, Sergey Bereg, and Sergio Cabello. They let me share part of their wisdom and experience in Computational Geometry and Algorithms. I also give thanks to everyone else whom I have worked with in open problem workshops. I would like to show my gratitude with the invitations of: Jorge Urrutia to the UNAM, Sergio Cabello to the University of Ljubljana, Clemens Hue- mer to the Universitat Polit`ecnica de Catalunya, and Leslie Bajuelos to the University of Aveiro.
    [Show full text]
  • The NP-Hardness of Covering Points with Lines, Paths and Tours and Their Tractability with FPT-Algorithms
    The NP-Hardness of Covering Points with Lines, Paths and Tours and their Tractability with FPT-Algorithms Author Heednacram, Apichat Published 2010 Thesis Type Thesis (PhD Doctorate) School School of Information and Communication Technology DOI https://doi.org/10.25904/1912/1694 Copyright Statement The author owns the copyright in this thesis, unless stated otherwise. Downloaded from http://hdl.handle.net/10072/367754 Griffith Research Online https://research-repository.griffith.edu.au The NP-Hardness of Covering Points with Lines, Paths and Tours and their Tractability with FPT-Algorithms by Apichat Heednacram B. Eng. (1st Class Hons), Griffith University, 2001 Submitted in fulfilment of the requirements of the degree of Doctor of Philosophy Institute for Integrated and Intelligent Systems Science, Environment, Engineering and Technology Griffith University March 2010 i Abstract Given a problem for which no polynomial-time algorithm is likely to exist, we investigate how to attack this seemingly intractable problem based on parame- terized complexity theory. We study hard geometric problems, and show that they are fixed-parameter tractable (FPT) given an instance and a parameter k. This allows the problems to be solved exactly, rather than approximately, in polynomial time in the size of the input and exponential time in the parameter. Although the parameterized approach is still young, in recent years, there have been many results published concerning graph problems and databases. However, not many earlier results apply the parameterized approach in the field of computational geometry. This thesis, therefore, focuses on geometric NP-hard problems. These problems are the Line Cover problem, the Rectilinear Line Cover problem in higher dimensions, the Rectilinear Minimum-Links Spanning Path problem in higher dimensions, the Rectilinear Hyper- plane Cover problem, the Minimum-Bends Traveling Salesman Prob- lem and the Rectilinear Minimum-Bends Traveling Salesman Prob- lem, in addition to some other variants of these problems.
    [Show full text]
  • Integer Linear Programming Models for the Problem of Covering a Polygonal Region by Rectangles
    Integer Linear Programming Models for the Problem of Covering a Polygonal Region by Rectangles G. Scheithauer, Yu. Stoyan and T. Romanova Abstract– The aim of the paper is to develop in- require a lot of computational effort to prove that teger linear programming (ILP) models for the circumstance. problem of covering a polygonal region by rectan- gles. We formulate a Beasley-type model in which In [8] one-dimensional bar relaxations for the CPR the number of variables depends on the size pa- problem are proposed to be used as necessary condi- rameters. Another ILP model is proposed which tions for existence of coverings. has O(n2 max{m, n}) variables where m is the num- Covering problems are of interest in many fields of ber of edges of the target set and n is the number of given rectangles. In particular we consider the application. There are many relations between cov- case where the polygonal region is convex. Exten- ering and packing or cutting problems. For an an- sions are also discussed where we allow the polyg- notated survey on Cutting and Packing we refer to onal region to be a union of a finite number of [5]. Covering problems arise naturally in a variety of convex subsets. applications. For a comprehensive overview we refer to [4]. Index terms – Covering, Integer Linear Program- ming, Mathematical Modelling, Optimization As an example, query optimization in spatial databases is a source of covering problems. In this setting a query may correspond to a geometric re- Introduction and Problem For- gion and be phrased in a generic form using geomet- ric parameters.
    [Show full text]
  • Rectangle Blanket Problem: Binary Integer Linear Programming Formulation and Solution Algorithms
    Rectangle Blanket Problem: Binary integer linear programming formulation and solution algorithms Barıs¸Evrim Demiroz¨ a, I.˙ Kuban Altınelb, Lale Akaruna aDepartment of Computer Engineering, Bo˘gazi¸ciUniversity, 34342, Bebek, Istanbul,˙ Turkey bDepartment of Industrial Engineering, Bo˘gazi¸ciUniversity, 34342, Bebek, Istanbul,˙ Turkey Abstract A rectangle blanket is a set of non-overlapping axis-aligned rectangles, used to approximately represent the two- dimensional image of a shape approximately. The use of a rectangle blanket is a widely considered strategy for speeding-up the computations in many computer vision applications. Since neither the rectangles nor the image have to be fully covered by the other, the blanket becomes more precise as the non-overlapping area of the image and the blanket decreases. In this work, we focus on the rectangle blanket problem, which involves the determination of an optimum blanket minimizing the non-overlapping area with a given image subject to an upper bound on the total number of rectangles the blanket can include. This problem has similarities with rectangle covering, rectangle partitioning and cutting / packing problems. The image replaces an irregular master object by an approximating set of smaller axis-aligned rectangles. The union of these rectangles, namely, the rectangle blanket, is neither restricted to remain entirely within the master object, nor required to cover the master object completely. We first develop a binary integer linear programming formulation of the problem. Then, we introduce four methods for its solution. The first one is a branch-and-price algorithm that computes an exact optimal solution. The second one is a new constrained simulated annealing heuristic.
    [Show full text]
  • Point Visibility Graphs and Restricted-Orientation Polygon Covering / by David Dylan Bremner
    Point Visibility Graphs and Restricted-Orientation Polygon Covering David Dylan Bremner B.Sc. (Hons.), University of Calgary, 1990 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TBE DEGREE OF MASTEROF SCIENCE in the School of Computing Science @ David Dylan Bremner 1993 SIMON FRASER UNIVERSITY April 1993 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: David Dylan Bremner Degree: Master of Science Title of thesis: Point Visibility Graphs and Restricted-Orient ation Polygon Covering Examining Committee: Dr. Lou Hafer Chair Dr. Tbomas Shermer , Senior Supervisor Assistant Professor of Computing Science Dr. Binay Bhattacharya Professor ofComputing Science Dr. Pavol Hell Professor of Computing Science Dr. Robin Dawes External Examiner Assistant Professor of Computer Science Queen's University Date Approved: PARTIAL COPYRIGHT LICENSE I hereby grant to Simon Fraser University the right to lend my thesis, project or extended essay (the title of which is shown below) to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the l i brary of any other university, or other educational institution, on its own behalf or for one of its users. I further agree that permission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for flnanclal gain shal I not be allowed without my written permlsslon.
    [Show full text]
  • Computational and Communication Complexity of Geometric Problems
    Computational and Communication Complexity of Geometric Problems by Sima Hajiaghaei Shanjani B.Sc. Amirkabir University of Technology, 2011 B.Sc. Amirkabir University of Technology, 2012 M.Sc. Sharif University of Technology, 2015 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Computer Science c Sima Hajiaghaei Shanjani, 2021 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. We acknowledge with respect the Lekwungen peoples on whose traditional territory the university stands and the Songhees, Esquimalt and WSÁNEĆ peoples whose ¯ historical relationships with the land continue to this day. ii Computational and Communication Complexity of Geometric Problems by Sima Hajiaghaei Shanjani B.Sc. Amirkabir University of Technology, 2011 B.Sc. Amirkabir University of Technology, 2012 M.Sc. Sharif University of Technology, 2015 Supervisory Committee Dr. Valerie King, Supervisor (Department of Computer Science) Dr. Venkatesh Srinivasan, Departmental Member (Department of Computer Science) Dr. William Evans, Outside Member (Department of Computer Science, University of British Columbia ) iii ABSTRACT In this dissertation, we investigate a number of geometric problems in different settings. We present lower bounds and approximation algorithms for geometric prob- lems in sequential and distributed settings. For the sequential setting, we prove the first hardness of approximation results for the following problems: • Red-Blue Geometric Set Cover is APX-hard when the objects are axis-aligned rectangles. c • Red-Blue Geometric Set Cover cannot be approximated to within 2log1−1/(log log m) m in polynomial time for any constant c < 1/2, unless P = NP , when the given objects are m triangles or convex objects.
    [Show full text]