Iterative Methods in Combinatorial Optimization
With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and si...
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| Médium: | E-kniha Kniha |
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Cambridge ; New York
Cambridge University Press
18.04.2011
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| Vydání: | 1 |
| Edice: | Cambridge Texts in Applied Mathematics |
| Témata: | |
| ISBN: | 9780521189439, 1107007518, 0521189438, 9781107007512 |
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| Abstract | With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms. |
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| AbstractList | With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. |
| Author | Lau, Lap Chi Singh, Mohit Ravi, R. |
| Author_xml | – sequence: 1 givenname: Lap Chi surname: Lau fullname: Lau, Lap Chi organization: The Chinese University of Hong Kong – sequence: 2 givenname: R. surname: Ravi fullname: Ravi, R. organization: Carnegie Mellon University, Pennsylvania – sequence: 3 givenname: Mohit surname: Singh fullname: Singh, Mohit organization: McGill University, Montréal |
| BackLink | https://cir.nii.ac.jp/crid/1130000797307228160$$DView record in CiNii |
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| Copyright | Lap Chi Lau, R. Ravi, and Mohit Singh 2011 |
| Copyright_xml | – notice: Lap Chi Lau, R. Ravi, and Mohit Singh 2011 |
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| Notes | Includes bibliographical references (p. 233-240) and index |
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| PageCount | 256 |
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| Snippet | With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual... This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate... |
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| SubjectTerms | Combinatorial optimization COMPUTERS / General bisacsh Iterative methods (Mathematics) |
| TableOfContents | Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Audience -- History -- Acknowledgments -- Dedications -- 1 Introduction -- 1.1 The assignment problem -- 1.2 Iterative algorithm -- 1.2.1 Contradiction proof idea: Lower bound > -- upper bound -- 1.2.2 Approximation algorithms for NP-hard problems -- 1.3 Approach outline -- 1.4 Context and applications of iterative rofinding -- 1.5 Book chapters overview -- 1.6 Notes -- 2 Preliminaries -- 2.1 Linear programming -- 2.1.1 Extreme point solutions to linear programs -- 2.1.1.1 Basic feasible solution -- 2.1.2 Algorithms for linear programming -- 2.1.3 Separation and optimization -- 2.1.4 Linear programming duality -- 2.1.4.1 Complementary slackness conditions -- 2.2 Graphs and digraphs -- 2.3 Submodular and supermodular functions -- 2.3.1 Submodularity -- 2.3.2 Supermodularity -- 2.3.3 Refinements -- 2.3.3.1 Minimizing submodular function -- Exercises -- 3 Matching and vertex cover in bipartite graphs -- 3.1 Matchings in bipartite graphs -- 3.1.1 Linear programming relaxation -- 3.1.2 Characterization of extreme point solutions -- 3.1.3 Iterative algorithm -- 3.1.4 Correctness and optimality -- 3.2 Generalized assignment problem -- 3.2.1 Linear programming relaxation -- 3.2.2 Characterization of extreme point solutions -- 3.2.3 Iterative algorithm -- 3.2.4 Correctness and performance guarantee -- 3.3 Maximum budgeted allocation -- 3.3.1 Linear programming relaxation -- 3.3.2 Characterization of extreme point solutions -- 3.3.3 An iterative 2-approximation algorithm -- 3.3.4 An iterative -approximation algorithm -- 3.3.5 Correctness and performance guarantee -- 3.4 Vertex cover in bipartite graphs -- 3.4.1 Linear programming relaxation -- 3.4.2 Characterization of extreme point solutions -- 3.4.3 Iterative algorithm -- 3.4.4 Correctness and optimality 8.4.2 Matroid intersection -- 8.4.3 Submodular flows -- 8.5 Notes -- Exercises -- 9 Matchings -- 9.1 Graph matching -- 9.1.1 Linear programming relaxation -- 9.1.2 Characterization of extreme point solutions -- 9.1.3 Iterative algorithm -- 9.1.4 Correctness and optimality -- 9.2 Hypergraph matching -- 9.2.1 Linear programming relaxation -- 9.2.2 Characterization of extreme point solutions -- 9.2.3 Iterative algorithm and local ratio method -- 9.2.4 Partial Latin square -- 9.3 Notes -- Exercises -- 10 Network design -- 10.1 Survivable network design problem -- 10.1.1 Linear programming relaxation -- 10.1.2 Characterization of extreme point solutions -- 10.1.3 Iterative algorithm -- 10.1.4 Correctness and performance guarantee -- 10.2 Connection to the traveling salesman problem -- 10.2.1 Linear programming relaxation -- 10.2.2 Characterization of extreme point solutions -- 10.2.3 Existence of edges with large fractional value -- 10.3 Minimum bounded degree Steiner networks -- 10.3.1 Linear programming relaxation -- 10.3.2 Characterization of extreme point solutions -- 10.3.3 Iterative algorithm -- 10.3.4 Correctness and performance guarantee -- 10.4 An additive approximation algorithm -- 10.4.1 Iterative algorithm -- 10.4.2 Correctness and performance guarantee -- 10.4.3 Steiner forests -- 10.5 Notes -- Exercises -- 11 Constrained optimization problems -- 11.1 Vertex cover -- 11.1.1 Linear programming relaxation -- 11.1.2 Characterization of extreme point solutions -- 11.1.3 Iterative algorithm -- 11.1.4 Correctness and performance guarantee -- 11.2 Partial vertex cover -- 11.2.1 Linear programming relaxation -- 11.2.2 Characterization of extreme point solutions -- 11.2.3 Iterative algorithm -- 11.2.4 Correctness and performance guarantee -- 11.3 Multicriteria spanning trees -- 11.3.1 Linear programming relaxation 3.5 Vertex cover and matching: duality -- 3.6 Notes -- Exercises -- 4 Spanning trees -- 4.1 Minimum spanning trees -- 4.1.1 Linear Programming Relaxation -- 4.1.2 Characterization of extreme point solutions -- 4.1.3 Uncrossing technique -- 4.1.4 Leaf-finding iterative algorithm -- 4.1.5 Correctness and optimality of leaf-finding algorithm -- 4.2 Iterative 1-edge-finding algorithm -- 4.2.1 Correctness and optimality of 1-edge-finding algorithm -- 4.3 Minimum bounded-degree spanning trees -- 4.3.1 Linear programming relaxation -- 4.3.2 Characterization of extreme point solutions -- 4.3.3 Leaf-finding iterative algorithm -- 4.3.4 Correctness and performance guarantee -- 4.4 An additive one approximation algorithm -- 4.4.1 Correctness and performance guarantee -- 4.5 Notes -- Exercises -- 5 Matroids -- 5.1 Preliminaries -- 5.2 Maximum weight basis -- 5.2.1 Linear programming formulation -- 5.2.2 Characterization of extreme point solutions -- 5.2.3 Iterative algorithm -- 5.2.4 Correctness and optimality -- 5.3 Matroid intersection -- 5.3.1 Linear programming relaxation -- 5.3.2 Characterization of extreme point solutions -- 5.3.3 Iterative algorithm -- 5.3.4 Correctness and optimality -- 5.4 Duality and min-max theorem -- 5.4.1 Dual of maximum weight basis -- 5.4.2 Dual of two matroid intersection -- 5.5 Minimum bounded degree matroid basis -- 5.5.1 Linear programming relaxation -- 5.5.2 Characterization of extreme point solutions -- 5.5.3 Iterative algorithm -- 5.5.4 Correctness and performance guarantee -- 5.5.5 Applications -- 5.6 k matroid intersection -- 5.6.1 Linear programming relaxation -- 5.6.2 Characterization of extreme point solutions -- 5.6.3 Iterative algorithm -- 5.6.4 Correctness and performance guarantee -- 5.7 Notes -- Exercises -- 6 Arborescence and rooted connectivity -- 6.1 Minimum cost arborescence 13.6.1 Linear programming relaxation 11.3.2 Characterization of extreme point solutions -- 11.3.3 Iterative algorithm -- 11.3.4 Correctness and performance guarantee -- 11.4 Notes -- Exercises -- 12 Cut problems -- 12.1 Triangle cover -- 12.1.1 Linear programming relaxation -- 12.1.2 Iterative algorithm -- 12.1.3 Correctness and performance guarantee -- 12.2 Feedback vertex set on bipartite tournaments -- 12.2.1 Linear programming relaxation -- 12.2.2 Iterative algorithm -- 12.2.3 Correctness and performance guarantee -- 12.3 Node multiway cut -- 12.3.1 Linear programming relaxation -- 12.3.2 Iterative algorithm -- 12.3.3 Correctness and performance guarantee -- 12.4 Notes -- Exercises -- 13 Iterative relaxation: Early and recent examples -- 13.1 A discrepancy theorem -- 13.1.1 Linear programming relaxation -- 13.1.2 Characterization of extreme point solutions -- 13.1.3 Iterative algorithm -- 13.1.4 Correctness and performance guarantee -- 13.2 Rearrangments of sums -- 13.2.1 Linear programming relaxation -- 13.2.2 Characterization of extreme point solutions -- 13.2.3 Iterative algorithm -- 13.2.4 Correctness and performance guarantee -- 13.3 Minimum cost circulation -- 13.3.1 Linear programming relaxation -- 13.3.2 Characterization of extreme point solutions -- 13.3.3 Iterative algorithm -- 13.3.4 Correctness and optimalit -- 13.4 Minimum cost unsplittable flow -- 13.4.1 Linear programming relaxation -- 13.4.2 Iterative algorithm -- 13.4.3 Correctness and performance guarantee -- 13.5 Bin packing -- 13.5.1 Linear programming relaxation -- 13.5.2 Characterization of extreme point solutions -- 13.5.3 Defining residual problems: Grouping and elimination -- 13.5.3.1 Linear grouping -- 13.5.3.2 Geometric grouping -- 13.5.3.3 Elimination of small items -- 13.5.4 Iterative algorithm -- 13.5.5 Correctness and performance guarantee -- 13.6 Iterative randomized rofinding: Steiner trees 6.1.1 Linear programming relaxation -- 6.1.2 Characterization of extreme point solutions -- 6.1.3 Iterative algorithm -- 6.1.4 Correctness and optimality -- 6.2 Minimum cost rooted k-connected subgraphs -- 6.2.1 Linear programming relaxation -- 6.2.2 Iterative algorithm -- 6.2.3 Characterization of extreme point solutions -- 6.2.4 Correctness and optimality -- 6.3 Minimum bounded degree arborescence -- 6.3.1 Linear programming relaxation -- 6.3.2 Characterization of extreme point solutions -- 6.3.3 Iterative algorithm -- 6.3.4 Correctness and performance guarantee -- 6.4 Additive performance guarantee -- 6.4.1 Iterative algorithm -- 6.4.2 Correctness and performance guarantee -- 6.5 Notes -- Exercises -- 7 Submodular flows and applications -- 7.1 The model and the main result -- 7.1.2 Generalizing to submodular functions -- 7.2 Primal integrality -- 7.2.2 Characterization of extreme point solutions -- 7.2.3 Iterative algorithm -- 7.2.4 Correctness and optimality -- 7.3 Dual integrality -- 7.4 Applications of submodular flows -- 7.4.1 Directed cut cover and feedback arc set -- 7.4.1.1 Feedback arc set indirected planar graphs -- 7.4.2 Polymatroid intersection -- 7.4.3 Graph orientation -- 7.5 Minimum bounded degree submodular flows -- 7.5.1 Linear Programming Relaxation -- 7.5.2 Characterization of extreme point solutions -- 7.5.3 Iterative algorithm -- 7.5.4 Correctness and performance guarantee -- 7.5.5 Applications -- 7.6 Notes -- Exercises -- 8 Network matrices -- 8.1 The model and main results -- 8.2 Primal integrality -- 8.2.1 Linear programming relaxation -- 8.2.3 Iterative algorithm -- 8.2.4 Correctness and optimality -- 8.3 Dual integrality -- 8.3.1 Linear programming relaxation -- 8.3.2 Characterization of extreme point solutions -- 8.3.3 Iterative algorithm -- 8.3.4 Correctness and optimality -- 8.4 Applications -- 8.4.1 Matroid |
| Title | Iterative Methods in Combinatorial Optimization |
| URI | http://dx.doi.org/10.1017/CBO9780511977152 https://doi.org/10.1017/CBO9780511977152?locatt=mode:legacy https://cir.nii.ac.jp/crid/1130000797307228160 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=691988 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9780511977152 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781107221772&uid=none |
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