Computational complexity: a modern approach
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
[2009]
|
| Subjects: | |
| Online Access: | Volltext#Teil Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references and index |
| Physical Description: | xxiv, 579 Seiten Diagramme |
| ISBN: | 9780521424264 |
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MARC
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| 100 | 1 | |a Arora, Sanjeev |d 1968- |e Verfasser |0 (DE-588)113855516 |4 aut | |
| 245 | 1 | 0 | |a Computational complexity |b a modern approach |c Sanjeev Arora (Princeton University), Boaz Barak (Princeton University) |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c [2009] | |
| 264 | 4 | |c © 2009 | |
| 300 | |a xxiv, 579 Seiten |b Diagramme | ||
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Record in the Search Index
| _version_ | 1824928736731987968 |
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| adam_text |
Contents
About this book page
xiii
Acknowledgments
xvii
Introduction
xix
0
Notational conventions
. 1
0.1
Representing objects as strings
2
0.2
Decision problems/languages
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
I
.
I 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
Ρ
24
1.7
Proof of Theorem
1.9:
Universal simulation in O(T log T)-time
29
CHAPTER NOTES AND HISTORY
32
EXERCISES
34
2
NP and NP completeness
. 38
2.1
The Class NP
39
2.2
Reducibilitv 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
viii Contents
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
Σ^
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,,^
107
6.2
Uniformly generated circuits 111
6.3
Turing machines that take advice
112
6.4
P/p,,,,,
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
¡x
8 Interactive
proofs
. 143
8.1 Interactive
proofs: Some variations
144
8.2
Public coins and AM
150
8.3
IP
=
PSPACE
157
8.4
The power of the
prover
162
8.5 Multiprover
interactive proofs
(МІР)
163
8.6
Program checking
164
8.7
Interactive proof for the permanent
167
CHAPTER NOTES AND HISTORY
169
EXERCISES
170
9
Cryptography
. 172
9.1
Perfect secrecy and its limitations
173
9.2
Computational security, one-way functions, and pseudorandom generators
175
9.3
Pseudorandom generators from one-way permutations
180
9.4
Zero knowledge
186
9.5
Some applications
189
CHAPTER NOTES AND HISTORY
194
EXERCISES
197
10
Quantum computation
. 201
10.1
Quantum weirdness: The two-slit experiment
202
10.2
Quantum superposition and qubits
204
10.3
Definition of quantum computation and BQP
209
10.4
Grover's search algorithm
216
10.5
Simon's algorithm
219
10.6
Shor's algorithm: Integer factorization using quantum computers
221
10.7
BQP and classical complexity classes
230
CHAPTER NOTES AND HISTORY
232
EXERCISES
234
11
PCP
theorem and hardness of approximation: An introduction
. 237
11.1
Motivation: Approximate solutions to NP-hard optimization problems
238
11.2
Two views of the
PCP
Theorem
240
11.3
Equivalence of the two views
244
11.4
Hardness of approximation for vertex cover and independent set
247
11.5
NP ç
PCPfpolyW.l):
PCP
from the Walsh-Hadamard code
249
CHAPTER NOTES AND HISTORY
254
EXERCISES
255
PART TWO: LOWER BOUNDS FOR CONCRETE COMPUTATIONAL MODELS
257
12
Decision trees
. 259
12.1
Decision trees and decision tree complexity
259
12.2
Certificate complexity
262
12.3
Randomized decision trees
263
x
Contents
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
AC and Hastad'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
344
17.3
#P completeness
345
17.4
Toda's theorem: PH
с
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
di stnp
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
=
Ρ
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
. 42
1
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 2C
S
Рц.-:
Tradeoff between gap and alphabet size
472
22.4
Hâstad'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 qCSP instances into "nice" instances
491
CHAPTER NOTES AND HISTORY
493
EXERCISES
495
xü
Contents
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.I 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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| author | Arora, Sanjeev 1968- Barak, Boaz |
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| language | English |
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| publisher | Cambridge University Press |
| record_format | marc |
| spelling | Arora, Sanjeev 1968- Verfasser (DE-588)113855516 aut Computational complexity a modern approach Sanjeev Arora (Princeton University), Boaz Barak (Princeton University) Cambridge Cambridge University Press [2009] © 2009 xxiv, 579 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references and index Computational complexity Komplexitätstheorie (DE-588)4120591-1 gnd rswk-swf Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Komplexitätstheorie (DE-588)4120591-1 s DE-604 Berechnungskomplexität (DE-588)4134751-1 s 1\p DE-604 Barak, Boaz Verfasser (DE-588)13846023X aut Erscheint auch als Online-Ausgabe https://theory.cs.princeton.edu/complexity/ Verlag kostenfrei Volltext#Teil Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017557511&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Arora, Sanjeev 1968- Barak, Boaz Computational complexity a modern approach Computational complexity Komplexitätstheorie (DE-588)4120591-1 gnd Berechnungskomplexität (DE-588)4134751-1 gnd |
| subject_GND | (DE-588)4120591-1 (DE-588)4134751-1 |
| 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 (Princeton University), Boaz Barak (Princeton University) |
| title_fullStr | Computational complexity a modern approach Sanjeev Arora (Princeton University), Boaz Barak (Princeton University) |
| title_full_unstemmed | Computational complexity a modern approach Sanjeev Arora (Princeton University), Boaz Barak (Princeton University) |
| title_short | Computational complexity |
| title_sort | computational complexity a modern approach |
| title_sub | a modern approach |
| topic | Computational complexity Komplexitätstheorie (DE-588)4120591-1 gnd Berechnungskomplexität (DE-588)4134751-1 gnd |
| topic_facet | Computational complexity Komplexitätstheorie Berechnungskomplexität |
| url | https://theory.cs.princeton.edu/complexity/ http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017557511&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT arorasanjeev computationalcomplexityamodernapproach AT barakboaz computationalcomplexityamodernapproach |