CS278: Computational Complexity Spring 2001 Luca Trevisan These are scribed notes from a graduate course on Computational Complexity o®ered at the University of California at Berkeley in the Spring of 2001. The notes have been only minimally edited, and there may be several errors and imprecisions. I wish to thank the students who attended this course for their enthusiasm and hard work. Berkeley, June 12, 2001. Luca Trevisan 1 Contents 1 Introduction 3 2 Cook's Theorem 6 2.1 The Problem SAT . 6 2.2 Intuition for the Reduction . 7 2.3 The Reduction . 8 2.4 The NP-Completeness of 3SAT . 10 3 More NP-complete Problems 11 3.1 Independent Set is NP-Complete . 11 3.2 Nondeterministic Turing Machines . 12 3.3 Space-Bounded Complexity Classes . 13 4 Savitch's Theorem 15 4.1 Reductions in NL . 15 4.2 NL Completeness . 16 4.3 Savitch's Theorem . 17 4.4 coNL . 18 5 NL=coNL, and the Polynomial Hierarchy 19 5.1 NL = coNL . 19 5.1.1 A simpler problem ¯rst . 19 5.1.2 Finding r . 20 5.2 The polynomial hierarchy . 21 5.2.1 Stacks of quanti¯ers . 22 5.2.2 The hierarchy . 22 5.2.3 An alternate characterization . 23 6 Circuits 24 6.1 Circuits . 24 6.2 Relation to other complexity classes . 25 7 Probabilistic Complexity Classes 30 7.1 Probabilistic complexity classes . 30 7.2 Relations between complexity classes (or where do probabilistic classes stand?) 31 2 7.3 A BPPalgorithm for primality testing . 35 8 Probabilistic Algorithms 37 8.1 Branching Programs . 37 8.2 Testing Equivalence . 37 8.3 The Schwartz-Zippel Lemma . 39 9 BPP versus PH, and Counting Problems 41 9.1 BPP § . 41 ⊆ 2 9.2 Counting Classes . 42 10 Approximate Counting 44 10.1 Complexity of counting problems . 44 10.2 An approximate comparison procedure . 45 10.3 Constructing a-comp . 46 10.4 The proof of the Leftover Hash Lemma . 48 11 Valiant-Vazirani, and Introduction to Cyptography 49 11.1 Unique SAT and NP . 49 11.2 Cryptography . 52 12 One-way Functions and Pseudorandom Generators 54 12.1 Pseudorandom Generators and One-Way Functions . 54 12.2 The Connection between PRG and OWF . 55 13 Pseudorandom Generators and Pseudorandom Functions 60 13.1 A Pseudorandom generator . 60 13.2 Pseudorandom function families . 62 13.3 Cryptography . 64 14 Goldreich-Levin 67 14.1 Linear Boolean Functions on 0; 1 n . 67 f g 14.2 Error Correcting Codes (ECC) . 68 14.2.1 E±cient Decoding . 68 14.2.2 Dealing With Noisy Channels . 69 14.3 Hardcore Predicates . 71 15 Levin's Theory of Average-Case Complexity 73 15.1 Distributional Problems . 73 15.2 DistNP . 74 15.3 Polynomial-Time Samplability . 74 15.4 Reductions . 75 15.5 Polynomial-Time on Average . 75 15.6 Some Results . 76 15.7 Existence of Complete Problems . 76 3 16 Average-Case Complexity of the Permanent 78 16.1 The Permanent Problem . 78 17 Interactive Proofs 81 17.1 Interactive Proofs . 81 17.1.1 NP+ Interaction = NP . 82 17.1.2 NP+ Randomness . 83 17.2 IP . 83 17.3 An Example: Graph Non-Isomorphism . 85 18 IP=PSPACE 87 18.1 UNSAT IP . 87 ⊆ 18.2 A Proof System for #SAT . 89 19 IP=PSPACE 91 19.1 PSPACE-Complete Language: TQBF . 91 19.2 Arithmetization of TQBF . 91 19.2.1 Naive Solution . 92 19.2.2 Revised Solution . 93 19.3 The Interactive Protocol . 94 19.4 Analysis . 95 20 PCP and Hardness of Approximation 96 20.1 Probabilistically Checkable Proofs . 96 20.2 PCP and MAX-3SAT . 96 20.2.1 Approximability . 96 20.2.2 Inapproximability . 97 20.2.3 Tighter result . 98 20.3 Max Clique . 98 20.3.1 Approximability . 98 20.3.2 Inapproximability . 99 21 NP = PCP [log n, polylog n] 101 21.1 Overview . 101 21.2 Arithmetization of 3SAT . 102 21.2.1 First Try . 103 21.2.2 Second Try . 103 21.2.3 Bundling Polynomials into a Single Polynomial . 105 21.3 Low-degree Testing . 106 21.4 Polynomial Reconstruction Algorithm . 106 21.5 Summary . 107 22 Parallelization 109 22.1 Main Lemma . 109 22.2 Another Low-Degree Test . 109 22.3 Curves in m . 110 F 4 22.4 Components of the Proof . 110 22.5 Veri¯cation . 111 23 Parallelization 112 23.1 Veri¯er that accesses to a constant number of places . 112 23.1.1 The proof . 113 23.1.2 The verifying procedure . 113 23.2 Veri¯er that reads a constant number of bits . 115 24 NP in PCP[poly(n),1] 117 24.1 Picking an NP-complete problem for the reduction . 117 24.2 The prover-veri¯er interaction . 117 24.3 Proof that the veri¯er is not fooled . 118 25 Pseudorandomness and Derandomization 121 26 Nisan-Wigderson 124 26.1 Notation . 124 26.2 The main result . 124 26.3 Interlude: Combinatorial Design . 125 26.4 Construction . 126 26.5 Proof . 127 27 Extractors 130 27.1 Nisan-Wigderson Construction . 130 27.2 Extractors . 132 28 Extractors and Error-Correcting Codes 135 28.1 Extractors and Error-Correcting Codes . 135 28.2 Construction of Extractors . ..