PART Ⅰ:ALG？RITHMS 1
1 Problems and Algorithms 3
1.1 Graph reachability 3
1.2 Maximum flow and matching 8
1.3 The traveling salesman problem 13
1.4 Notes,references,and problems 14
2 Turing machines 19
2.1 Turing machine basics 19
2.2 Turing machines as algorithms 24
2.3 Turing machines with multiple strings 26
2.4 Linear speedup 32
2.5 Space bounds 34
2.6 Random access machines 36
2.7 Nondeterministic machines 45
2.8 Notes,references,and problems 51
3 Undecidability 57
3.1 Universal Turing machines 57
3.2 The halting problem 58
3.3 More undecidability 60
3.4 Notes,references,and problems 66
PART Ⅱ:LOGIC 71
4 Boolean logic 73
4.1 Boolean Expressions 76
4.2 Satisfiability and validity 76
4.3 Boolean functions and circuits 79
4.4 Notes,references,and problems 84
5 First-order logic 87
5.1 The syntax of first-order logic 87
5.2 Models 90
5.3 Valid expressions 95
5.4 Axioms and proofs 100
5.5 The completeness theorem 105
5.6 Consequences of the completeness theorem 110
5.7 Second-order logic 113
5.8 Notes,references,and problems 118
6 Undecidability in logic 123
6.1 Axioms for number theory 123
6.2 Computation as a number-theoretic concept 127
6.3 Undecidability and incompleteness 131
6.4 Notes,references,and problems 135
PART Ⅲ:P AND NP 137
7 Relations between complexity classes 139
7.1 Complexity classes 139
7.2 The hierarchy theorem 143
7.3 The reachability method 146
7.4 Notes,references,and problems 154
8 Reductions and completeness 159
8.1 Reductions 159
8.2 Completeness 165
8.3 Logical characterizations 172
8.4 Notes,references,and problems 177
9 NP-complete problems 181
9.1 Problems in NP 181
9.2 Variants of satisfiability 183
9.3 Graph-theoretic problems 188
9.4 Sets and numbers 199
9.5 Notes,references,and problems 207
10 coNP and function problems 219
10.1 NP and coNP 219
10.2 Primality 222
10.3 Function problems 227
10.4 Notes,references,and problems 235
11 Randomized computation 241
11.1 Randomized algorithms 241
11.2 Randomized complexity classes 253
11.3 Random sources 259
11.4 Circuit complexity 267
11.5 Notes,references,and problems 272
12 Cryptography 279
12.1 One-way functions 279
12.2 Protocols 287
12.3 Notes,references,and problems 294
13 Approximability 299
13.1 Approximation algorithms 299
13.2 Approximation and complexity 309
13.3 Nonapproximability 319
13.4 Notes,references,and problems 323
14 On P vs.NP 329
14.1 The map of NP 329
14.2 Isomorphism and density 332
14.3 Oracles 339
14.4 Monot.one circuits 343
14.5 Notes,references,and problems 350
PART Ⅳ:INSIDE P 357
15 Parallel computation 359
15.1 Parallel algorithms 359
15.2 Parallel models of computation 369
15.3 The class NC 375
15.4 RNC algorithms 381
15.5 Notes,references,and problems 385
16 Logarithmic space 395
16.1 The L？NL problem 395
16.2 Alternation 399
16.3 Undirected reachability 401
16.4 Notes,references,and problems 405
PART Ⅴ:BEYOND NP 409
17 The polynomial hierarchy 411
17.1 Optimization problems 411
17.2 The hierarchy 424
17.3 Notes,references,and problems 433
18 Computation that counts 439
18.1 The permanent 439
18.2 The class ⊕P 447
18.3 Notes,references,and problems 452
19 Polynomial space 455
19.1 Alternation and games 455
19.2 Games against nature and interactive protocols 469
19.3 More PSPACE-complete problems 480
19.4 Notes,references,and problems 487
20 A glimpse beyond 491
20.1 Exponential time 491
20.2 Notes,references,and problems 499
Index 509
Author index 519