Cambridge University Press 978-0-521-42426-4 - Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak Index More information

Index

Note: The page numbers in bold indicate the location where the term is defined.

⊕SAT problem, see Computational Boolean, 107, 215 problem Counting lower bound, 115 #CYCLE problem, see Computational DC uniform, 119 problem Equivalence to straight-line programs, #SAT problem, see Computational 109 problem Uniform generation, 111 #P-completeness, 346 Lower bounds, 286–304, 415, 499 Decision version, 158 Quantum, 214 2-slit experiment, 202 Universal basis, 216 3COL problem CKT-SAT problem, 110 NP-completeness, 66 NP-completeness, 111 3SAT problem, see Computational problem Clique problem 3SAT problem NP-completeness, 65 Average case, 363 CNF form, 44 Coin tossing, 192 Adleman’s theorem, 135 Combinatorial auction problem Algorithmica, see Impagliazzo’s worlds NP-completeness, 66 Alternating Turing machines, see Turing Combinatorial design, 410, 412 machine Communication complexity, 271 Approximation algorithm, 238 Computability theory, 21 Arithmetization, 158 Computational problem Average-case hardness, see Hardness ⊕SAT, 353, 355 Averaging principle, 147, 310, 377, 409, 512 #CYCLE, 342, 343 #SAT, 342 Baker-Gill-Solovay theorem, 74 3COL problem Bell’s inequalities, 207 NP-completeness, 66 Berman’s Theorem, 66 3SAT,44 Big-Oh notation, 3 3SAT problem Blum-Shub-Smale model, see Turing Average case, 363 Machine CIRCUIT-EVAL problem P-completeness, 119 Chebychev’s inequality, 184 CKT-SAT problem, 110 Chernoff bound, 133 Exactly-one 3SAT Church-Turing thesis, 26 NP-completeness, 65 CIRCUIT-EVAL problem Clique problem P-completeness, 119 NP-completeness, 65 Circuits Combinatorial auction problem Algebraic, 322 NP-completeness, 66

575

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576 Index

Computational problem (cont.) Decision trees, 260 Constraint satisfaction, 244 Algebraic, 326 Disjointness, 273 Designs, see Combinatorial designs PATH, 82 Determinant, 122, 130, 325, 346, 523 PATH problem Discrepancy, 275 NL-completeness Disjointness problem, see Computational Factoring, 40, 221 problem Graph connectivity, 25, 40 Distributional problems, 364 Graph isomorphism, 40, 148, 156, 165, 188, 242 Easy witness method, 416 Halting, 23 Eigenvalue, 422–426, 526 Hamiltonian cycle problem Eigenvector, see Eigenvalue NP-completeness, 65 Einstein-Podosky-Rosen paradox, 206 Encryption Independent set, 39 Perfect secrecy, 175 Integer programming, 40 Private key, 178 Integer programming problem Public key, 178 NP-completeness, 52 Error correcting code, 379 Linear programming, 40 Concatenated, 384 Max-cut problem Gilbert-Varshamov bound, 380 NP-completeness, 65 List decoding, 392 Perfect matching, 130 Local decoding, 386 Permanent, 166–169, 242, 323, Long code, 480 325, 414 Reed-Muller, 383 Permanent Local decoder, 387 #P-completeness, 364–351 Local list decoding, 395 Planted clique, 362 Reed-Solomon, 382 Polynomial identity testing, 129, 403 Berlekamp-Welsh decoder, 385 Primality testing, 40, 64, 128 Sudan list decoder, 394 Quadratic equations problem Walsh-Hadamard, 212, 249–250, 382 NP-completeness, 65 Local decoder, 250, 388 Quadratic residuosity, 149 Local list decoding, 183, 395 SAT, 38, 44, 44–50, 54–56, 62, 65, 71–73, Local testing, 477 77–78, 83–84, 92–95, 101–104, 106, Error reduction, 132, 431 113–114, 158, 167, 216–217, 231, 242, Exactly-one 3SAT 254–255, 315, 319, 333, 341, 352, 354, NP-completeness, 65 369–370, 419 Expander graphs, 426–431 Set cover, 486 Chernoff bound, 434 Error reduction, 431 2SAT,67 Subset sum, 40 Explicit construction, 434–440 Subset sum problem Use in PCP, 463–470, 491 Expander mixing lemma, 429 NP-completeness, 65 Extractors, 442–453 Subset sum problem pseudo-polynomial time algorithm, 66 Factoring problem, see Computational Tautology, 56 problem TQBF, 161, 165 Feasable interpolation theorem, 312 Traveling salesperson, 40 Fingerprinting, 130 Unique SAT, 354 Fourier transform, 223 UPATH, 139, 426, 440 GF(2)n, 475 Configuration graph, 80 Cyclic groups, 223 Constraint satisfaction problem, Fast Fourier transform, xxi, 223 see Computational problem Quantum, 224 Continued fractions, 229 Frege proof system, 315 Cook reductions, see Reductions Cook-Levin theorem, 45, 110 Gödel’s Theorem, 23 Quantum analog, 231 Goldreich-Levin theorem, 183, 395 Cryptomania, see Impagliazzo’s worlds Alternative proof, 497 Cutting planes proof system, 313 Goldwasser-Sipser theorem, 151

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Index 577

Graph connectivity problem, Log rank conjecture, 283 see Computational problem Logical independence, 75 Graph isomorphism problem, see Computational problem, Machine learning, 193 Grover’s algorithm, 216 Mandelbrot set, 333 Max-cut problem Hadamard matrix, see Error correcting code NP-completeness, 65 Halting problem, see Computational problem Metric spaces, 527 Hamiltonian cycle problem Meyer’s theorem, 114, 416 NP-completeness, 65 Min-max theorem, 379 Hamiltonian path problem Minicrypt, see Impagliazzo’s worlds NP-completeness, 53 Multi-party secure computation, 192 Hardcore lemma, see Impagliazzo’s hardcore lemma Natural proofs, 499 Hardness Negligible function, 176 Average-case, 375, 406 Nisan-Wigderson Generator, Worst-case, 375 see Pseudorandom generator Håstad’s PCP theorem, 474 NP-completeness, 42 Håstad’s switching lemma, 287 Nullstellensatz, 60, 333 Heuristica, see Impagliazzo’s worlds Proof system, 314 Hierarchy theorem deterministic time, 69 non-deterministic time, 69 One-time pad, 175 Non-uniform, 116 One-way function Probabilistic?, 138 Rabin, 177 space, 82 RSA, 177 Hilbert’s Nullstellensatz, see Nullstellensatz Universal, 178 Oracle Turing Machines, see Turing Immerman-Szelepcsényi theorem, 91 machine Impagliazzo’s hardcore lemma, 376 Impagliazzo’s worlds P-computable distributions, 365 Algorithmica, 369 P-sampleable distributions, 365, 369 Cryptomania, 370 Pairwise independence, 396, 514 Heuristica, 369 Extension to k > 2, 536 Minicrypt, 370 Goldreich-Levin theorem, 184 Pessiland, 369 Hash functions, 152, 354, 445 Independent set problem, see Computational Leftover hash lemma, 445 problem Parallel repetition theorem, 473 Hardness of approximation, 247 PATH problem, see Computational problem NP-completeness, 51 PATH problem Integer programming problem, NL-completeness, 89 NP-completeness, 52 PCP theorem, 241, 461 Interactive proofs, 146 Perfect matching problem, see Computational Multi-prover proofs, 163 problem IP = PSPACE, 158 Permanent, 166–169, 242, 323, 325, 414 Ising model, 343 #P-completeness, 346–351 Pessiland, see Impagliazzo’s worlds Karp reductions, see Reductions Planted clique problem, see Computational Karp-Lipton theorem, 113 problem k-wise independent hash functions, Polynomial calculus proof system, 314 see Pairwise independence Polynomial identity testing problem, see Computational problem Ladner’s theorem, 71 Primality testing problem, see Computational Lattices, 196, 233, 364 problem, Leftover hash lemma, see Pairwise Program checking, 164 independence Pseudorandom functions, 189, 503 Levin reductions, see Reductions Pseudorandom generator Linear programming problem, see Derandomization, 404 Computational problem From one-way functions, 180

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578 Index

Pseudorandom generator (cont.) Straight-line program From one-way permutations, 180 Algebraic, 319 Log-space, 449 Boolean, 107 Nisan-Wigderson, 410 Subset sum problem, Secure, 180 NP-completeness, 65 Uniform, 413 Pseudo-polynomial time algorithm, 66 Yao’s theorem, 181 Sumcheck protocol, 159 PSPACE, 158 Tautology problem, see Computational Quadratic equations problem problem NP-completeness, 65 Tensor product, 251, 284, 435 Quadratic residuosity problem, Time constructible functions, 16 see Computational problem Time/space tradeoffs for SAT, 101 Quantum register, see Qubit Toda’s theorem, 352 Qubit, 204, 210 TQBF problem, see Computational problem TQBF problem, 84 Traveling salesperson problem, Rabin encryption, see One-way function see Computational problem Rabinovitch’s trick, 330 Turing machine, 12 Random self reducibility, 166 Advice, 112 Random subsum principle, 249 Alternating, 99 Random walk, 139, 422–426 Blum-Shub-Smale model, 331 Razborov-Smolensky theorem, 291 Non-deterministic, 41 Real quadratic equations problem Non-deterministic universal, 64 NP-completeness, 66 oblivious, 18 Reductions Oracle, 72 Average-case, 366 Probabilistic, 125 Cook, 65 single tape, 17 Karp, 42 Space-bounded, 78 Levin, 50, 248 Universal, 20 log-space, 88 Universal space-bounded, 93 Parsimonious, 65 Randomized, 138 Unary languages, 66 Search to decision, 55 Unique games conjecture, 490 Replacement product, 436 Unique SAT, 354 Resolution, 309 Unitary matrices, 209 Reversible computation, 211 UPATH problem, see Computational problem Rotation maps, 434 RSA encryption, see One-way function Valiant-Vazirani theorem, 217, 354 Vertex cover problem SAT problem, see Computational problem 2-approximation, 67 Savitch’s theorem, 86 Hardness of approximation, 247 Search to decision reduction, see Reductions NP-completeness, 65 Sensitivity of functions, 266 Set cover problem, see Computational Weak random sources, 133, 442 problem Worst-case hardness, see Hardness Set lower bound protocol, 152 Shannon secrecy, see Encryption XOR Lemma, see Yao’s XOR Lemma Shor’s algorithm, 221 2SAT problem, see Computational problem Yao’s min-max lemma, 265 Simon’s algorithm, 219 Yao’s XOR Lemma, 375 Sipser-Gács theorem, 136 Space hierarchy theorem, see Hierarchy Zero knowledge, 187 theorem Zig-zag product, 440

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