Theory of Computational Complexity

Preface ix Notes on the Second Edition xv Part I Uniform Complexity 1 1 Models of Computation and Complexity Classes 3 1.1 Strings, Coding, and Boolean Functions 3 1.2 Deterministic Turing Machines 7 1.3 Nondeterministic Turing Machines 14 1.4 Complexity Classes 17 1.5 Universal Turing Machine 23 1.6 Diagonalization 27 1.7 Simulation 31 Exercises 35 Historical Notes 41 2 NP-Completeness 43 2.1 NP 43 2.2 Cook's Theorem 47 2.3 More NP-Complete Problems 51 2.4 Polynomial-Time Turing Reducibility 58 2.5 NP-Complete Optimization Problems 64 Exercises 71 Historical Notes 75 3 The Polynomial-Time Hierarchy and Polynomial Space 77 3.1 Nondeterministic Oracle Turing Machines 77 3.2 Polynomial-Time Hierarchy 79 3.3 Complete Problems in PH 84 3.4 Alternating Turing Machines 90 3.5 PSPACE-Complete Problems 95 3.6 EXP-Complete Problems 102 Exercises 108 Historical Notes 111 4 Structure of NP 113 4.1 Incomplete Problems in NP 113 4.2 One-Way Functions and Cryptography 116 4.3 Relativization 122 4.4 Unrelativizable Proof Techniques 124 4.5 Independence Results 125 4.6 Positive Relativization 126 4.7 Random Oracles 128 4.8 Structure of Relativized NP 132 Exercises 137 Historical Notes 140 Part II Nonuniform Complexity 141 5 Decision Trees 143 5.1 Graphs and Decision Trees 143 5.2 Examples 149 5.3 Algebraic Criterion 153 5.4 Monotone Graph Properties 157 5.5 Topological Criterion 159 5.6 Applications of the Fixed Point Theorems 166 5.7 Applications of Permutation Groups 169 5.8 Randomized Decision Trees 172 5.9 Branching Programs 177 Exercises 184 Historical Notes 188 6 Circuit Complexity 191 6.1 Boolean Circuits 191 6.2 Polynomial-Size Circuits 195 6.3 Monotone Circuits 201 6.4 Circuits with Modulo Gates 208 6.5 NC 212 6.6 Parity Function 217 6.7 P-Completeness 224 6.8 Random Circuits and RNC 230 Exercises 234 Historical Notes 237 7 Polynomial-Time Isomorphism 241 7.1 Polynomial-Time Isomorphism 241 7.2 Paddability 245 7.3 Density of NP-Complete Sets 250 7.4 Density of EXP-Complete Sets 258 7.5 One-Way Functions and Isomorphism in EXP 262 7.6 Density of P-Complete Sets 272 Exercises 275 Historical Notes 278 Part III Probabilistic Complexity 281 8 Probabilistic Machines and Complexity Classes 283 8.1 Randomized Algorithms 283 8.2 Probabilistic Turing Machines 288 8.3 Time Complexity of Probabilistic Turing Machines 291 8.4 Probabilistic Machines with Bounded Errors 294 8.5 BPP and P 297 8.6 BPP and NP 300 8.7 BPP and the Polynomial-Time Hierarchy 302 8.8 Relativized Probabilistic Complexity Classes 306 Exercises 311 Historical Notes 314 9 Complexity of Counting 317 9.1 Counting Class #P 318 9.2 #P-Complete Problems 321 9.3 oplus P and the Polynomial-Time Hierarchy 330 9.4 #P and the Polynomial-Time Hierarchy 336 9.5 Circuit Complexity and Relativized oplus P and #P 338 9.6 Relativized Polynomial-Time Hierarchy 342 Exercises 344 Historical Notes 347 10 Interactive Proof Systems 349 10.1 Examples and Definitions 349 10.2 Arthur-Merlin Proof Systems 357 10.3 AM Hierarchy Versus Polynomial-Time Hierarchy 361 10.4 IP Versus AM 368 10.5 IP Versus PSPACE 378 Exercises 383 Historical Notes 386 11 Probabilistically Checkable Proofs and NP-Hard Optimization Problems 389 11.1 Probabilistically Checkable Proofs 389 11.2 PCP Characterization of NP 392 11.2.1 Expanders 396 11.2.2 Gap Amplification 399 11.2.3 Assignment Testers 410 11.4 Probabilistic Checking and Inapproximability 418 11.5 More NP-Hard Approximation Problems 421 Exercises 432 Historical Notes 435 Bibliography 439 Index 461