Contents
PART I: ALGORITHMS
Problems and Algorithms
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
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
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
4 Boolean logic
4.1 Boolean Expressions
4.2 Satisfiability and validity 76
4.3 Boolean functions and circuits 79
4.4 Notes, references, and problems 84
5 First-order logic
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
6.1 Axioms for number theory 123
6.2 Computation as a number-theoretic concept
6.3 Undecidability and incompleteness 131
6.4 Notes, references, and problems 135
PART III: P AND NP
7 Relations between complexity classes
7.1 Complexity classes 139
7.2 The hierarchy theorem 143
7.3 The teachability method 146
7.4 Notes, references, and problems 154
8 Reductions and completeness
8.1 Reductions 159
8.2 Completeness 165
Contents
8.3 Logical characterizations 172
8.4 Notes, references, and problems 177
9 NP-complete problems
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
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
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
12.1 One-way functions 279
12.2 Protocols 287
12.3 Notes, references, and problems 294
13 Approximability
13.1 Approximation algorithms 299
13.2 Approximation and complexity 309
13.3 Nonapproximability 319
13.4 Notes, references, and problems 323
On P vs. NP
14.1 The map of NP 329
14.2 Isomorphism and density 332
14.3 Oracles 339
14.4 Monotone circuits 343
14.5 Notes, references, and problems 350
PART IV: INSIDE P
15 Parallel computation
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
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
17 The polynomial hierarchy
17.1 Optimization problems 411
17.2 The hierarchy 424
17.3 Notes, references, and problems 433
18 Computation that counts
18.1 The permanent 439
18.2 The class iBP 447
18.3 Notes, references, and problems 452
Contents
19 Polynomial space
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
20.1 Exponential time 491
20.2 Notes, references, and problems 499
Index
Author index
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