Computational complexity: a modern approach
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2010
|
| Edition: | Reprint |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XXIV, 579 Seiten Illustrationen, Diagramme |
| ISBN: | 9780521424264 |
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Record in the Search Index
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| adam_text | Contents
About this book page ism
Acknowledgments xvii
Introduction xix
0 Notational conventions................ . . . . . . V ; . . . 1
0.1 Representing objects as strings 2
0.2 Decision problems/Ianguages 3
0.3 Big-oh notation 3
exercises 4
PART ONE: BASIC COMPLEXITY CLASSES • - 7
1 The computational model—and why It doesn t matter . ....... . . 9
1.1 Modeling computation: What you really need to know 10
1.2 The Turing machine 11
1.3 Efficiency and running time 15
1.4 Machines as strings and the universal Turing machine 19
1.5 Uncomputability: An introduction 21
1.6 The Class P 24
1.7 Proof of Theorem 1.9: Universal simulation in 0(T log 7 time 29
CHAPTER NOTES AND HISTORY • - 32
EXERCISES 34
2 NP and NP completeness . .......... ... . ........... 38
2.1 The Class NP 39
2.2 Reducibility and NP-completeness 42
2.3 The Cook-Levin Theorem: Computation is local 44
2.4 The web of reductions 50
2.5 Decision versus search 54
2.6 coNP, EXP, and NEXP 55
2.7 More thoughts about P, NP, and all that 57
CHAPTER NOTES AND HISTORY 62
EXERCISES 63
vii
Contents
vm
3 Diagonalization.......................................................... 68
3.1 Time Hierarchy Theorem 69
3.2 Nondeterministic Time Hierarchy Theorem 69
3.3 Ladner s Theorem: Existence of NP-intermediate problems 71
3.4 Oracle machines and the limits of diagonalization 72
CHAPTER NOTES AND HISTORY 76
EXERCISES 77
4 Space complexity . ....................................................... 78
4.1 Definition of space-bounded computation 78
4.2 PSPACE completeness 83
4.3 NL completeness 87
CHAPTER NOTES AND HISTORY 93
EXERCISES 93
5 The polynomial hierarchy and alternations................................ 95
5.1 The Class 96
5.2 The polynomial hierarchy 97
5.3 Alternating Turing machines 99
5.4 Time versus alternations: Time-space tradeoffs for SAT 101
5.5 Defining the hierarchy via oracle machines 102
CHAPTER NOTES AND HISTORY 104
EXERCISES 104
6 Boolean circuits ....................................................... 106
6.1 Boolean circuits and P/p()|y 107
6.2 Uniformly generated circuits 111
6.3 Turing machines that take advice 112
6.4 P/po,y and NP 113
6.5 Circuit lower bounds 115
6.6 Nonuniform Hierarchy Theorem 116
6.7 Finer gradations among circuit classes 116
6.8 Circuits of exponential size 119
CHAPTER NOTES AND HISTORY 120
EXERCISES 121
7 Randomized computation................................................ 123
7.1 Probabilistic Turing machines 124
7.2 Some examples of PTMs 126
7.3 One-sided and “zero-sided” error: RP, coRP, ZPP 131
7.4 The robustness of our definitions 132
7.5 Relationship between BPP and other classes 135
7.6 Randomized reductions 138
7.7 Randomized space-bounded computation 139
CHAPTER NOTES AND HISTORY 140
EXERCISES 141
Contents
8 Interactive proofs . ........v. ..... . . .... ... . . .
8.1 Interactive proofs: Some variations
8.2 Public coins and AM
8 3 IP = PSPACE
8.4 The power of the prover
8.5 Multiprover interactive proofs (MIP)
8.6 Program checking
8.7 Interactive proof for the permanent
CHAPTER NOTES AND HISTORY
EXERCISES
9 Cryptography...............................................
9.1 Perfect secrecy and its limitations
9.2 Computational security, one-way functions, and pseudorandom generators
9.3 Pseudorandom generators from one-way permutations
9.4 Zero knowledge
9.5 Some applications
CHAPTER NOTES AND HISTORY /
EXERCISES
10 Quantum computation ....................... . . . .
10.1 Quantum weirdness: The two-slit experiment
10.2 Quantum superposition and qubits
10.3 Definition of quantum computation and BQP
10.4 Grover’s search algorithm
10.5 Simon’s algorithm
10.6 Shor’s algorithm: Integer factorization using quantum computers
10.7 BQP and classical complexity classes
CHAPTER NOTES AND HISTORY
EXERCISES
11 PCP theorem and hardness of approximation: An introduction . . .....
j rjr11.1 Motivation: Approximate solutions to NP-hard optimization problems
11.2 Two views of the PCP Theorem
11.3 Equivalence of the two views
11.4 Hardness of approximation for vertex cover and independent set
11.5 NP c PCP(poly(«),l): PCP from the Walsh-Hadamard code ,
CHAPTER NOTES AND HISTORY
I EXERCISES
PARTTWO: LOWER BOUNDS FOR CONCRETE COMPUTATIONAL MODELS
12
Decision trees ...... . . . . . . ...
12.1 Decision trees and decision tree complexity
12.2 Certificate complexity
12.3 Randomized decision trees
ix
143
144
150
157
162
163
164
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169
170
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186
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244
247
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254
255
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259
262
263
Contents
x
12.4 Some techniques for proving decision tree lower bounds 264
CHAPTER NOTES AND HISTORY 268
EXERCISES 269
13 Communication complexity . . . ... .. . ... .... ....... . . 270
13.1 Definition of two-party communication complexity 271
13.2 Lower bound methods 272
13.3 Multiparty communication complexity 278
13.4 Overview of other communication models 280
CHAPTER NOTES AND HISTORY 282
EXERCISES 283
14 Circuit lower bounds: Complexity theory s Waterloo . . ............... 286
14.1 AC0 and Hàstad’s Switching Lemma 286
14.2 Circuits with “counters”: ACC 291
14.3 Lower bounds for monotone circuits 293
14.4 Circuit complexity: The frontier 297
14.5 Approaches using communication complexity 300
CHAPTER NOTES AND HISTORY 304
EXERCISES 305
15 Proof complexity ._______ ....... ...... . ...... . ... . . 307
15.1 Some examples 307
15.2 Propositional calculus and resolution 309
15.3 Other proof systems: A tour d’horizon 313
15.4 Metamathematical musings 315
CHAPTER NOTES AND HISTORY 316
EXERCISES 317
16 Algebraic computation models ......................................... 318
16.1 Algebraic straight-line programs and algebraic circuits 319
16.2 Algebraic computation trees 326
16.3 The Blum-Shub-Smale model 331
CHAPTER NOTES AND HISTORY 334
EXERCISES 336
PART THREE: ADVANCED TOPICS 339
17 Complexity of counting . . . ... . . . ............ ........... . . . . 341
17.1 Examples of counting problems 342
17.2, The Class #P V. ; 344
17.3 #P completeness 345
17.4 Toda’s theorem: PII ç P#SAT 352
17.5 Open problems 358
CHAPTER NOTES AND HISTORY 359
EXERCISES 359
Contents
xi
18 Average case complexity: Levin s theory . .... . ...... ...... 361
18.1 Distributional problems and distP 362
18.2 Formalization of “real-life distributions ’ 365
18.3 distnp and its complete problems 365
18.4 Philosophical and practical implications 369
CHAPTER NOTES AND HISTORY 371
EXERCISES 371
19 Hardness amplification and error-correcting codes.......................... 373
19.1 Mild to strong hardness: Yao’s XOR lemma 375
19.2 Tool: Error-correcting codes 379
19.3 Efficient decoding 385
19.4 Local decoding and hardness amplification 386
19.5 List decoding 392
19.6 Local list decoding: Getting to BPP = P 394
CHAPTER NOTES AND HISTORY 398
EXERCISES 399
20 Derandomization . . . ................................................... . 402
20.1 Pseudorandom generators and derandomization 403
20.2 Proof of Theorem 20.6: Nisan-Wigderson Construction 407
20.3 Derandomization under uniform assumptions 413
20.4 Derandomization requires circuit lower bounds 415
CHAPTER NOTES AND HISTORY 418
EXERCISES 419
21 Pseudorandom constructions: Expanders and extractors ...................... 421
21.1 Random walks and eigenvalues 422
21.2 Expander graphs 426
21.3 Explicit construction of expander graphs 434
21.4 Deterministic logspace algorithm for undirected connectivity 440
21.5 Weak random sources and extractors 442
21.6 Pseudorandom generators for space-bounded computation 449
CHAPTER NOTES AND HISTORY 454
EXERCISES 456
22 Proofs of PCP theorems and the Fourier transform technique . . . . . . 460
22.1 Constraint satisfaction problems with nonbinary alphabet 461
22.2 Proof of the PCP theorem 461
22.3 Hardness of 2CSPyv-Tradeoff between gap and alphabet size 472
22.4 H stad’s 3-bit PCP Theorem and hardness of MAX-3SAT 474
22.5 Tool: The Fourier transform technique 475
22.6 Coordinate functions, long Code, and its testing 480
22.7 Proof of Theorem 22.16 481
22.8 Hardness of approximating SET-COVER 486
• 22.9 Other PCP theorems: A survey 488
22.A Transforming tfCSP instances into “nice” instances 491
CHAPTER NOTES AND HISTORY 493
EXERCISES 495
Contents
xu
23 Why are circuit lower bounds so difficult? . . ..... ... . . . . . . 498
23.1 Definition of natural proofs 499
23.2 What s so natural about natural proofs? 500
23.3 Proof of Theorem 23.1 503
23.4 An “unnatural” lower bound 504
23.5 A philosophical view 505
CHAPTER NOTES AND HISTORY 506
EXERCISES 507
Appendix: Mathematical background ... . . ..................... . . . ..... . . 508
A.l Sets, functions, pairs, strings, graphs, logic 509
A.2 Probability theory 510
A.3 Number theory and groups 517
A.4 Finite fields 521
A.5 Basic facts from linear Algebra 522
A.6 Polynomials 527
Hints and selected exercises 531
Main theorems and definitions 545
Bibliography 549
Index 575
Complexity class index 579
|
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| spelling | Arora, Sanjeev 1968- Verfasser (DE-588)113855516 aut Computational complexity a modern approach Sanjeev Arora ; Boaz Barak Reprint Cambridge [u.a.] Cambridge Univ. Press 2010 XXIV, 579 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Computational complexity Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Komplexitätstheorie (DE-588)4120591-1 gnd rswk-swf Komplexitätstheorie (DE-588)4120591-1 s DE-604 Berechnungskomplexität (DE-588)4134751-1 s DE-188 Barak, Boaz Verfasser (DE-588)13846023X aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022635350&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arora, Sanjeev 1968- Barak, Boaz Computational complexity a modern approach Computational complexity Berechnungskomplexität (DE-588)4134751-1 gnd Komplexitätstheorie (DE-588)4120591-1 gnd |
| subject_GND | (DE-588)4134751-1 (DE-588)4120591-1 |
| title | Computational complexity a modern approach |
| title_auth | Computational complexity a modern approach |
| title_exact_search | Computational complexity a modern approach |
| title_full | Computational complexity a modern approach Sanjeev Arora ; Boaz Barak |
| title_fullStr | Computational complexity a modern approach Sanjeev Arora ; Boaz Barak |
| title_full_unstemmed | Computational complexity a modern approach Sanjeev Arora ; Boaz Barak |
| title_short | Computational complexity |
| title_sort | computational complexity a modern approach |
| title_sub | a modern approach |
| topic | Computational complexity Berechnungskomplexität (DE-588)4134751-1 gnd Komplexitätstheorie (DE-588)4120591-1 gnd |
| topic_facet | Computational complexity Berechnungskomplexität Komplexitätstheorie |
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