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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. 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 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. S0 | a subset of S; that is, S0 S. ⊆ n0 | the size of S0 where n0 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. d e d e Lk+1 | the set of all lines that cover at least k + 1 points in S. L | the cardinality of a set L. j j 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=pN) 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.
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