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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| Hlavní autori: | , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Cambridge ; New York
Cambridge University Press
18.04.2011
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| Vydanie: | 1 |
| Edícia: | Cambridge Texts in Applied Mathematics |
| Predmet: | |
| ISBN: | 9780521189439, 1107007518, 0521189438, 9781107007512 |
| On-line prístup: | Získať plný text |
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- 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