Lecture Notes on Computational Complexity Luca Trevisan1 Notes written in Fall 2002, Revised May 2004 1Computer Science Division, U.C. Berkeley. Work supported by NSF Grant No. 9984703. Email [email protected]. Foreword These are scribed notes from a graduate courses on Computational Complexity o®ered at the University of California at Berkeley in the Fall of 2002, based on notes scribed by students in Spring 2001 and on additional notes scribed in Fall 2002. I added notes and references in May 2004. The ¯rst 15 lectures cover fundamental material. The remaining lectures cover more advanced material, that is di®erent each time I teach the class. In these Fall 2002 notes, there are lectures on Hºastad's optimal inapproximability results, lower bounds for parity in bounded depth-circuits, lower bounds in proof-complexity, and pseudorandom generators and extractors. The notes have been only minimally edited, and there may be several errors and impre- cisions. I will be happy to receive comments, criticism and corrections about these notes. The syllabus for the course was developed jointly with Sanjeev Arora. Sanjeev wrote the notes on Yao's XOR Lemma (Lecture 11). Many people, and especially Avi Wigderson, were kind enough to answer my questions about various topics covered in these notes. I wish to thank the scribes Scott Aaronson, Andrej Bogdanov, Allison Coates, Kama- lika Chaudhuri, Steve Chien, Jittat Fakcharoenphol, Chris Harrelson, Iordanis Kerenidis, Lawrence Ip, Ranjit Jhala, Fran»cois Labelle, John Leen, Michael Manapat, Manikandan Narayanan, Mark Pillo®, Vinayak Prabhu, Samantha Riesenfeld, Stephen Sorkin, Kunal Talwar, Jason Waddle, Dror Weitz, Beini Zhou. Berkeley, May 2004. Luca Trevisan [email protected] 1 Contents 1 Introduction, P and NP 7 1.1 Computational Problems . 7 1.2 P and NP . 8 1.3 NP-completeness . 9 1.3.1 Reductions . 9 1.3.2 NP-completeness . 9 1.3.3 An NP-complete problem . 10 1.3.4 The Problem SAT . 10 1.4 Diagonalization . 11 1.5 References . 12 2 Space-Bounded Complexity Classes 14 2.1 Space-Bounded Complexity Classes . 14 2.2 Reductions in NL . 15 2.3 NL Completeness . 16 2.4 Savitch's Theorem . 17 2.5 Undirected Connectivity . 18 2.6 Randomized Log-space . 18 2.7 NL = coNL . 20 2.7.1 A simpler problem ¯rst . 20 2.7.2 Finding r . 21 3 The Polynomial Hierarchy 23 3.1 Stacks of quanti¯ers . 23 3.2 The hierarchy . 24 3.3 An alternate characterization . 24 3.4 Additional Properties . 25 3.5 References . 26 4 Circuits 28 4.1 Circuits . 28 4.2 Relation to other complexity classes . 29 4.3 References . 33 2 5 Probabilistic Complexity Classes 35 5.1 Probabilistic complexity classes . 35 5.2 Relations between complexity classes . 36 5.3 BPP § . 40 ⊆ 2 5.4 References . 41 6 A Probabilistic Algorithm 43 6.1 Branching Programs . 43 6.2 Testing Equivalence . 44 6.3 The Schwartz-Zippel Lemma . 45 6.4 References . 45 7 Unique-SAT 47 7.1 The Valiant-Vazirani Reduction . 47 7.2 References . 49 8 Counting Problems 51 8.1 Counting Classes . 51 8.2 Complexity of counting problems . 52 8.3 An approximate comparison procedure . 53 8.4 Constructing a-comp . 54 8.5 The proof of the Leftover Hash Lemma . 55 8.6 Approximate Sampling . 56 8.7 References . 57 9 Average-Case Complexity of the Permanent 58 9.1 The Permanent Problem . 58 9.2 Worst-case to Average-case Equivalence . 59 9.3 References . 60 10 Average-case Complexity of Problems in PSPACE and EXP 61 10.1 Average-case Complexity in PSPACE and EXP for 7/8 Fraction of Inputs 61 10.2 Coding-Theoretic Perspective . 62 10.3 Average-case Complexity in PSPACE and EXP for 1/2 Fraction of Inputs 64 10.4 References . 66 11 The XOR Lemma 68 11.1 The XOR Lemma . 68 11.2 References . 70 12 Levin's Theory of Average-case Complexity 72 12.1 Distributional Problems . 72 12.2 DistNP . 73 12.3 Reductions . 73 12.4 Polynomial-Time on Average . 74 12.5 Existence of Complete Problems . 77 3 12.6 Polynomial-Time Samplability . 78 12.7 References . 79 13 Interactive Proofs 81 13.1 Interactive Proofs . 81 13.1.1 NP+ Interaction = NP . 82 13.1.2 NP + Randomness . 83 13.2 IP . 83 13.3 An Example: Graph Non-Isomorphism . 84 13.4 Random Self-Reductions . 85 13.5 References . 87 14 IP=PSPACE 89 14.1 UNSAT IP . 89 ⊆ 14.2 A Proof System for #SAT . 91 14.3 A Proof System for QBF . 92 14.3.1 PSPACE-Complete Language: TQBF . 92 14.3.2 Arithmetization of TQBF . 92 14.3.3 The Interactive Protocol . 95 14.3.4 Analysis . 96 15 Introduction to PCP 98 15.1 Probabilistically Checkable Proofs . 98 15.2 PCP and MAX-3SAT . 98 15.2.1 Approximability . 98 15.2.2 Inapproximability . 99 15.2.3 Tighter result . 100 15.3 Max Clique . 100 15.3.1 Approximability . 100 15.3.2 Inapproximability . 101 15.4 References . 101 16 Hºastad's Proof System 102 16.1 Hºastad's veri¯er . 102 16.2 Applications to hardness of approximation . 103 16.3 From the PCP theorem to Raz's veri¯er . 105 16.4 References . 106 17 Linearity Test 107 17.1 Fourier analysis of Boolean functions . 107 17.2 Using Fourier analysis to test linearity . 110 17.3 Testing code words for the Long Code . 111 17.3.1 Pre-Hºastad veri¯cation . 111 17.3.2 The Long Code test . 112 17.4 References . 112 4 18 Testing the Long Code 114 18.1 Test for Long Code . 114 18.2 Hºastad's veri¯er . 115 19 Circuit Lower Bounds for Parity Using Polynomials 119 19.1 Circuit Lower Bounds for PARITY . 119 19.2 Proof of Circuit Lower Bounds for PARITY using Polynomials . ..