Approximation algorithms:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York
Springer
2003
|
| Edition: | Corrected second printing |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 380 S. Illustrationen, Diagramme |
| ISBN: | 3540653678 9783642084690 9783540653677 |
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| 100 | 1 | |a Vazirani, Vijay V. |d 1957- |e Verfasser |0 (DE-588)122932196 |4 aut | |
| 245 | 1 | 0 | |a Approximation algorithms |c Vijay V. Vazirani |
| 250 | |a Corrected second printing | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York |b Springer |c 2003 | |
| 300 | |a XIX, 380 S. |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Algorithmes | |
| 650 | 7 | |a Algoritmos |2 larpcal | |
| 650 | 7 | |a Combinatória |2 larpcal | |
| 650 | 4 | |a Optimisation mathématique | |
| 650 | 7 | |a Otimização combinatória |2 larpcal | |
| 650 | 4 | |a Computer algorithms | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 0 | 7 | |a Approximationsalgorithmus |0 (DE-588)4500954-5 |2 gnd |9 rswk-swf |
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Table of Contents
1 Introduction 1
1.1 Lower bounding OPT 2
1.1.1 An approximation algorithm for cardinality vertex cover 3
1.1.2 Can the approximation guarantee be improved? 3
1.2 Well-characterized problems and min-max relations 5
1.3 Exercises 7
1.4 Notes 10
Part I. Combinatorial Algorithms
2 Set Cover 15
2.1 The greedy algorithm 16
2.2 Layering 17
2.3 Application to shortest superstring 19
2.4 Exercises 22
2.5 Notes 26
3 Steiner Tree and TSP 27
3.1 Metric Steiner tree 27
3.1.1 MST-based algorithm 28
3.2 Metric TSP 30
3.2.1 A simple factor 2 algorithm 31
3.2.2 Improving the factor to 3/2 32
3.3 Exercises 33
3.4 Notes 37
4 Multiway Cut and fc-Cut 38
4.1 The multiway cut problem 38
4.2 The minimum fc-cut problem 40
4.3 Exercises 44
4.4 Notes 46
XIV Table of Contents
5 fc-Center 47
5.1 Parametric pruning applied to metric fc-center 47
5.2 The weighted version 50
5.3 Exercises 52
5.4 Notes 53
6 Feedback Vertex Set 54
6.1 Cyclomatic weighted graphs 54
6.2 Layering applied to feedback vertex set 57
6.3 Exercises 60
6.4 Notes 60
7 Shortest Superstring 61
7.1 A factor 4 algorithm 61
7.2 Improving to factor 3 64
7.2.1 Achieving half the optimal compression 66
7.3 Exercises 66
7.4 Notes 67
8 Knapsack 68
8.1 A pseudo-polynomial time algorithm for knapsack 69
8.2 An FPTAS for knapsack 69
8.3 Strong NP-hardness and the existence of FPTAS's 71
8.3.1 Is an FPTAS the most desirable approximation
algorithm? 72
8.4 Exercises 72
8.5 Notes 73
9 Bin Packing 74
9.1 An asymptotic PTAS 74
9.2 Exercises 77
9.3 Notes 78
10 Minimum Makespan Scheduling 79
10.1 Factor 2 algorithm 79
10.2 A PTAS for minimum makespan 80
10.2.1 Bin packing with fixed number of object sizes 81
10.2.2 Reducing makespan to restricted bin packing 81
10.3 Exercises 83
10.4 Notes 83
11 Euclidean TSP 84
11.1 The algorithm 84
11.2 Proof of correctness 87
11.3 Exercises 89
11.4 Notes 89
Table of Contents XV
Part II. LP-Based Algorithms
12 Introduction to LP-Duality 93
12.1 The LP-duality theorem 93
12.2 Min-max relations and LP-duality 97
12.3 Two fundamental algorithm design techniques 100
12.3.1 A comparison of the techniques and the notion of
integrality gap 101
12.4 Exercises 103
12.5 Notes 107
13 Set Cover via Dual Fitting 108
13.1 Dual-fitting-based analysis for the greedy set cover algorithm 108
13.1.1 Can the approximation guarantee be improved? Ill
13.2 Generalizations of set cover 112
13.2.1 Dual fitting applied to constrained set multicover 112
13.3 Exercises 116
13.4 Notes 117
14 Rounding Applied to Set Cover 118
14.1 A simple rounding algorithm 118
14.2 Randomized rounding 119
14.3 Half-integrality of vertex cover 121
14.4 Exercises 122
14.5 Notes 123
15 Set Cover via the Primal-Dual Schema 124
15.1 Overview of the schema 124
15.2 Primal-dual schema applied to set cover 126
15.3 Exercises 128
15.4 Notes 129
16 Maximum Satisfiability 130
16.1 Dealing with large clauses 131
16.2 Derandomizing via the method of conditional expectation . 131
16.3 Dealing with small clauses via LP-rounding 133
16.4 A 3/4 factor algorithm 135
16.5 Exercises 136
16.6 Notes 138
17 Scheduling on Unrelated Parallel Machines 139
17.1 Parametric pruning in an LP setting 139
17.2 Properties of extreme point solutions 140
17.3 The algorithm 141
XVI Table of Contents
17.4 Additional properties of extreme point solutions 142
17.5 Exercises 143
17.6 Notes 144
18 Multicut and Integer Multicommodity Flow in Trees 145
18.1 The problems and their LP-relaxations 145
18.2 Primal-dual schema based algorithm 148
18.3 Exercises 151
18.4 Notes 153
19 Multiway Cut 154
19.1 An interesting LP-relaxation 154
19.2 Randomized rounding algorithm 156
19.3 Half-integrality of node multiway cut 159
19.4 Exercises 162
19.5 Notes 166
20 Multicut in General Graphs 167
20.1 Sum multicommodity flow 167
20.2 LP-rounding-based algorithm 169
20.2.1 Growing a region: the continuous process 170
20.2.2 The discrete process 171
20.2.3 Finding successive regions 172
20.3 A tight example 174
20.4 Some applications of multicut 175
20.5 Exercises 176
20.6 Notes 178
21 Sparsest Cut 179
21.1 Demands multicommodity flow 179
21.2 Linear programming formulation 180
21.3 Metrics, cut packings, and ^-embeddability 182
21.3.1 Cut packings for metrics 182
21.3.2 ^-embeddability of metrics 184
21.4 Low distortion ^-embeddings for metrics 185
21.4.1 Ensuring that a single edge is not overshrunk 186
21.4.2 Ensuring that no edge is overshrunk 189
21.5 LP-rounding-based algorithm 190
21.6 Applications 191
21.6.1 Edge expansion 191
21.6.2 Conductance 191
21.6.3 Balanced cut 192
21.6.4 Minimum cut linear arrangement 193
21.7 Exercises 194
21.8 Notes 196
Table of Contents XVII
22 Steiner Forest 197
22.1 LP-relaxation and dual 197
22.2 Primal-dual schema with synchronization 198
22.3 Analysis 203
22.4 Exercises 206
22.5 Notes 211
23 Steiner Network 212
23.1 LP-relaxation and half-integrality 212
23.2 The technique of iterated rounding 216
23.3 Characterizing extreme point solutions 218
23.4 A counting argument 220
23.5 Exercises 223
23.6 Notes 230
24 Facility Location 231
24.1 An intuitive understanding of the dual 232
24.2 Relaxing primal complementary slackness conditions 233
24.3 Primal-dual schema based algorithm 234
24.4 Analysis 235
24.4.1 Running time 237
24.4.2 Tight example 237
24.5 Exercises 238
24.6 Notes 241
25 fc-Median 242
25.1 LP-relaxation and dual 242
25.2 The high-level idea 243
25.3 Randomized rounding 246
25.3.1 Derandomization 247
25.3.2 Running time 248
25.3.3 Tight example 248
25.3.4 Integrality gap 249
25.4 A Lagrangian relaxation technique
for approximation algorithms 249
25.5 Exercises 250
25.6 Notes 253
26 Semidefinite Programming 255
26.1 Strict quadratic programs and vector programs 255
26.2 Properties of positive semidefinite matrices 257
26.3 The semidefinite programming problem 258
26.4 Randomized rounding algorithm 260
26.5 Improving the guarantee for MAX-2SAT 263
26.6 Exercises 265
26.7 Notes 268
XVIII Table of Contents
Part III. Other Topics
27 Shortest Vector 273
27.1 Bases, determinants, and orthogonality defect 274
27.2 The algorithms of Euclid and Gauss 276
27.3 Lower bounding OPT using Gram-Schmidt orthogonalization 278
27.4 Extension to n dimensions 280
27.5 The dual lattice and its algorithmic use 284
27.6 Exercises 288
27.7 Notes 292
28 Counting Problems 294
28.1 Counting DNF solutions 295
28.2 Network reliability 297
28.2.1 Upperbounding the number of near-minimum cuts. 298
28.2.2 Analysis 300
28.3 Exercises 302
28.4 Notes 305
29 Hardness of Approximation 306
29.1 Reductions, gaps, and hardness factors 306
29.2 The PCP theorem 309
29.3 Hardness of MAX-3SAT 311
29.4 Hardness of MAX-3SAT with bounded occurrence
of variables 313
29.5 Hardness of vertex cover and Steiner tree 316
29.6 Hardness of clique 318
29.7 Hardness of set cover 322
29.7.1 The two-prover one-round characterization of NP . 322
29.7.2 The gadget 324
29.7.3 Reducing error probability by parallel repetition 325
29.7.4 The reduction 326
29.8 Exercises 329
29.9 Notes 332
30 Open Problems 334
30.1 Problems having constant factor algorithms 334
30.2 Other optimization problems 336
30.3 Counting problems 338
30.4 Notes 343
Table of Contents XIX
Appendix
A An Overview of Complexity Theory
for the Algorithm Designer 344
A.I Certificates and the class NP 344
A.2 Reductions and NP-completeness 345
A. 3 NP-optimization problems and approximation algorithms . 346
A.3.1 Approximation factor preserving reductions 348
A.4 Randomized complexity classes 348
A.5 Self-reducibility 349
A.6 Notes 352
B Basic Facts from Probability Theory 353
B.I Expectation and moments 353
B.2 Deviations from the mean 354
B.3 Basic distributions 355
B.4 Notes 355
References 357
Problem Index 373
Subject Index 377 |
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| isbn | 3540653678 9783642084690 9783540653677 |
| language | English |
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| physical | XIX, 380 S. Illustrationen, Diagramme |
| publishDate | 2003 |
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| spelling | Vazirani, Vijay V. 1957- Verfasser (DE-588)122932196 aut Approximation algorithms Vijay V. Vazirani Corrected second printing Berlin ; Heidelberg ; New York Springer 2003 XIX, 380 S. Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Algorithmes Algoritmos larpcal Combinatória larpcal Optimisation mathématique Otimização combinatória larpcal Computer algorithms Mathematical optimization Approximationsalgorithmus (DE-588)4500954-5 gnd rswk-swf Approximationsalgorithmus (DE-588)4500954-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010544448&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vazirani, Vijay V. 1957- Approximation algorithms Algorithmes Algoritmos larpcal Combinatória larpcal Optimisation mathématique Otimização combinatória larpcal Computer algorithms Mathematical optimization Approximationsalgorithmus (DE-588)4500954-5 gnd |
| subject_GND | (DE-588)4500954-5 |
| title | Approximation algorithms |
| title_auth | Approximation algorithms |
| title_exact_search | Approximation algorithms |
| title_full | Approximation algorithms Vijay V. Vazirani |
| title_fullStr | Approximation algorithms Vijay V. Vazirani |
| title_full_unstemmed | Approximation algorithms Vijay V. Vazirani |
| title_short | Approximation algorithms |
| title_sort | approximation algorithms |
| topic | Algorithmes Algoritmos larpcal Combinatória larpcal Optimisation mathématique Otimização combinatória larpcal Computer algorithms Mathematical optimization Approximationsalgorithmus (DE-588)4500954-5 gnd |
| topic_facet | Algorithmes Algoritmos Combinatória Optimisation mathématique Otimização combinatória Computer algorithms Mathematical optimization Approximationsalgorithmus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010544448&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT vaziranivijayv approximationalgorithms |