Approximation Algorithms For Clustering Problems
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Approximation Algorithms For Clustering And Facility Location Problems
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Author : Sara Ahmadian
language : en
Publisher:
Release Date : 2017
Approximation Algorithms For Clustering And Facility Location Problems written by Sara Ahmadian and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Approximation theory categories.
Facility location problems arise in a wide range of applications such as plant or warehouse location problems, cache placement problems, and network design problems, and have been widely studied in Computer Science and Operations Research literature. These problems typically involve an underlying set F of facilities that provide service, and an underlying set D of clients that require service, which need to be assigned to facilities in a cost-effective fashion. This abstraction is quite versatile and also captures clustering problems, where one typically seeks to partition a set of data points into k clusters, for some given k, in a suitable way, which themselves find applications in data mining, machine learning, and bioinformatics. Basic variants of facility location problems are now relatively well-understood, but we have much-less understanding of more-sophisticated models that better model the real-world concerns. In this thesis, we focus on three models inspired by some real-world optimization scenarios. In Chapter 2, we consider mobile facility location (MFL) problem, wherein we seek to relocate a given set of facilities to destinations closer to the clients as to minimize the sum of facility-movement and client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first local-search based approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is (3+epsilon)-approximation for this problem for any constant epsilon > 0 using local search which improves the previous best guarantee of 8-approximation algorithm due to [34] based on LP-rounding. Our results extend to the weighted generalization wherein each facility i has a non-negative weight w_i and the movement cost for i is w_i times the distance traveled by i. In Chapter 3, we consider a facility-location problem that we call the minimum-load k-facility location (MLkFL), which abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. This problem was studied under the name of min-max star cover in [32,10], who (among other results) gave bicriteria approximation algorithms for MLkFL when F=D. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polytime approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. In Chapter 4, we consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We consider objective functions involving the radii of open facilities, where the radius of a facility i is the maximum distance between i and a client assigned to it. We consider two problems: minimizing the sum of the radii of the open facilities, which yields the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, and minimizing the maximum radius, which yields the lower-bounded k-supplier with outliers (LBkSupO) problem. We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version. These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.
Approximation Algorithms For Clustering Problems
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Author : Chaitanya Swamy
language : en
Publisher:
Release Date : 2004
Approximation Algorithms For Clustering Problems written by Chaitanya Swamy and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with categories.
Approximation Algorithms For Clustering Problems
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Author : Babak Behsaz
language : en
Publisher:
Release Date : 2012
Approximation Algorithms For Clustering Problems written by Babak Behsaz and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Approximation theory categories.
Approximation Algorithms For Np Hard Clustering Problems
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Author : Ramgopal Reddy Mettu
language : en
Publisher:
Release Date : 2002
Approximation Algorithms For Np Hard Clustering Problems written by Ramgopal Reddy Mettu and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Algorithms categories.
Given a set of n points and their pairwise distances, the goal of clustering is to partition the points into a "small" number of "related" sets. Clustering algorithms are used widely to manage, classify, and summarize many kinds of data. In this dissertation, we study the classic facility location and k-median problems in the context of clustering, and formulate and study a new optimization problem that we call the online median problem. For each of these problems, it is known to be NP-hard to compute a solution with cost less than a certain constant factor times the optimal cost. We give simple constant-factor approximation algorithms for the facility location, k-median, and online median problems with optimal or near-optimal time bounds. We also study distance functions that are "approximately" metric, and show that such distance functions allow us to obtain a faster online median algorithm and to generalize our analysis to other objective functions, such as that of the well-known k-means heuristic. Given n points, the associated interpoint distances and nonnegative point weights, and a nonnegative penalty for each point, the facility location problem asks us to identify a set of cluster centers so that the weighted average cluster radii and the sum of the cluster center penalties are both minimized. The k-median problem asks us to identify exactly k cluster centers while minimizing just the weighted average cluster radii. We give a simple greedy algorithm for the facility location problem that runs in O(n^2) time and produces a solution with cost at most 3 times optimal. For the k-median problem, we develop and make use of a sampling technique that we call "successive sampling," and give a randomized constant-factor approximation algorithm that runs in O(n(k+\log{n}+\log^2{n})) time. We also give an Omega(nk) lower bound on the running time of any randomized constant-factor approximation algorithm for the k-median problem that succeeds with even a negligible constant probability. In many settings, it is desirable to browse a given data set at differing levels of granularity (i.e., number of clusters). To address this concern, we formulate a generalization of the k-median problem that we call the online median problem. The online median problem asks us to compute an ordering of the points so that, over all i, when a prefix of length i is taken as a set of cluster centers, the weighted average radii of the induced clusters is minimized. We show that a natural generalization of the greedy strategy that we call "hierarchically greedy" yields an algorithm that produces an ordering such that every prefix of the ordering is within a constant factor of the associated optimal cost. Furthermore, our algorithm has a running time of Theta(n^2). Finally, we study the performance of our algorithms in practice. We present implementations of our k-median and online median algorithms; our experimental results indicate that our approximation algorithms may be useful in practice.
Algorithms For Clustering Problems
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Author : Moses Samson Charikar
language : en
Publisher:
Release Date : 2000
Algorithms For Clustering Problems written by Moses Samson Charikar and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with categories.
Handbook Of Approximation Algorithms And Metaheuristics
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Author : Teofilo F. Gonzalez
language : en
Publisher: CRC Press
Release Date : 2018-05-15
Handbook Of Approximation Algorithms And Metaheuristics written by Teofilo F. Gonzalez and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-15 with Computers categories.
Handbook of Approximation Algorithms and Metaheuristics, Second Edition reflects the tremendous growth in the field, over the past two decades. Through contributions from leading experts, this handbook provides a comprehensive introduction to the underlying theory and methodologies, as well as the various applications of approximation algorithms and metaheuristics. Volume 1 of this two-volume set deals primarily with methodologies and traditional applications. It includes restriction, relaxation, local ratio, approximation schemes, randomization, tabu search, evolutionary computation, local search, neural networks, and other metaheuristics. It also explores multi-objective optimization, reoptimization, sensitivity analysis, and stability. Traditional applications covered include: bin packing, multi-dimensional packing, Steiner trees, traveling salesperson, scheduling, and related problems. Volume 2 focuses on the contemporary and emerging applications of methodologies to problems in combinatorial optimization, computational geometry and graphs problems, as well as in large-scale and emerging application areas. It includes approximation algorithms and heuristics for clustering, networks (sensor and wireless), communication, bioinformatics search, streams, virtual communities, and more. About the Editor Teofilo F. Gonzalez is a professor emeritus of computer science at the University of California, Santa Barbara. He completed his Ph.D. in 1975 from the University of Minnesota. He taught at the University of Oklahoma, the Pennsylvania State University, and the University of Texas at Dallas, before joining the UCSB computer science faculty in 1984. He spent sabbatical leaves at the Monterrey Institute of Technology and Higher Education and Utrecht University. He is known for his highly cited pioneering research in the hardness of approximation; for his sublinear and best possible approximation algorithm for k-tMM clustering; for introducing the open-shop scheduling problem as well as algorithms for its solution that have found applications in numerous research areas; as well as for his research on problems in the areas of job scheduling, graph algorithms, computational geometry, message communication, wire routing, etc.
Approximation Algorithms For Clustering With Minimum Sum Of Radii Diameters And Squared Radii
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Author : Mahya Jamshidian
language : en
Publisher:
Release Date : 2022
Approximation Algorithms For Clustering With Minimum Sum Of Radii Diameters And Squared Radii written by Mahya Jamshidian and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Approximation algorithms categories.
In this study, we present an improved approximation algorithm for three related problems. In the Minimum Sum of Radii clustering problem (MSR), we aim to select k balls in a metric space to cover all points while minimizing the sum of the radii. In the Minimum Sum of Diameters clustering problem (MSD), we are to pick k clusters to cover all the points such that sum of diameters of all the clusters is minimized. At last, in the Minimum Sum of Squared Radii problem (MSSR), the goal is to choose k balls, similar to MSR. However in MSSR, the goal is to minimize the sum of squares of radii of the balls. We present a 3.389-approximation for MSR and a 6.546-approximation for MSD, improving over respective 3.504 and 7.008 developed by Charkar and Panigrahy (2001). In particular, our guarantee for MSD is better than twice our guarantee for MSR. In the case of MSSR, the best known approximation guarantee is 4 ·(540)^2 based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the t-Metric Multicover Problem. With our analysis, we get a 11.078-approximation algorithm for Minimum Sum of Squared Radii.
Approximation Algorithms For Clustering Streams And Large Data Sets
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Author : Liadan O'Callaghan
language : en
Publisher:
Release Date : 2003
Approximation Algorithms For Clustering Streams And Large Data Sets written by Liadan O'Callaghan and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with categories.
Approximation Algorithms For Clustering And Facility Location Problems
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Author : Shalmoli Gupta
language : en
Publisher:
Release Date : 2018
Approximation Algorithms For Clustering And Facility Location Problems written by Shalmoli Gupta and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.
Approximation Algorithms For Clustering To Minimize The Sum Of Diameters
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Author :
language : en
Publisher:
Release Date : 2000
Approximation Algorithms For Clustering To Minimize The Sum Of Diameters written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with categories.
We consider the problem of partitioning the nodes of a complete edge weighted graph into [kappa] clusters so as to minimize the sum of the diameters of the clusters. Since the problem is NP-complete, our focus is on the development of good approximation algorithms. When edge weights satisfy the triangle inequality, we present the first approximation algorithm for the problem. The approximation algorithm yields a solution that has no more than 10k clusters such the total diameter of these clusters is within a factor O(log (n/[kappa])) of the optimal value fork clusters, where n is the number of nodes in the complete graph. For any fixed [kappa], we present an approximation algorithm that produces [kappa] clusters whose total diameter is at most twice the optimal value. When the distances are not required to satisfy the triangle inequality, we show that, unless P = NP, for any [rho] ≥ 1, there is no polynomial time approximation algorithm that can provide a performance guarantee of [rho] even when the number of clusters is fixed at 3. Other results obtained include a polynomial time algorithm for the problem when the underlying graph is a tree with edge weights.