Date Topics Relevant Chapters in Textbooks 1/8 Introduction. Computational problems, models of computation, algorithms and their efficiency. What is complexity theory. Goldreich Chapter 1. Arora-Barak Chapter 1. 1/10 P and NP, and the P-vs-NP problem. Karp reductions and NP-Completeness. The existence of NP-Complete problems. The statement of the Cook-Levin Theorem: SAT is NP-Complete. Goldreich Chapter 2. Arora-Barak Chapters 1 - 2. 1/15 Some applications of the Cook-Levin Theorem and more NP-Complete problems. Nondeterministic Turing machines and an alternative definition for NP. Goldreich Chapter 2. Arora-Barak Chapter 2. 1/17 Proof of the Cook-Levin Theorem. Cook reductions, and decision-vs-search problems. On NP languages neither in P nor NP-Complete: Ladner's Theorem. Goldreich Chapter 2. Arora-Barak Chapters 2, 3. See also this exposition on Ladner's Theorem. 1/22 (Impagliazzo's) Proof of Ladner's Theorem. Arora-Barak Chapters 3. See also this exposition on Ladner's Theorem. 1/24 The Polynomial Hierarchy. Introducing Boolean circuits. Arora-Barak Chapter 5. Goldreich Chapter 3. 1/29 A nonuniform model of computation: Boolean Circuits and the complexity classes they define. A nonconstructive result: Almost all Boolean functions have high circuit complexity. P/poly, and the Karp-Lipton theorem. Arora-Barak Chapter 6. Goldreich Chapter 3. 1/31 More on circuits. Introducing space complexity. Arora-Barak Chapter 6, 4. Goldreich Chapter 3, 5. 2/5 Space Complexity: Basic relations between time and space. Determinism vs Nondeterminism. Savitch's Theorem. PSPACE. TQBF is PSPACE-Complete. Arora-Barak Chapter 4. Goldreich Chapter 5. 2/7 Space Complexity: TQBF is PSPACE-Complete. NL. STCONN is NL-Complete. The Immerman-Szlepcsenyi Theorem. Arora-Barak Chapter 4. Goldreich Chapter 5. 2/12 Proof of the Immerman-Szlepcsenyi Theorem. Space and Time Hierarchy Theorems. Arora-Barak Chapter 4, 3. Goldreich Chapter 5, 4. 2/14 Hierarchy Theorems. Limits of relativizing techniques: Baker, Gill, Solovay. Introducing randomness. Arora-Barak Chapter 3, 7. Goldreich Chapter 4, 6. 2/19, 2/21, 2/26 Randomness in computation. Some randomized complexity classes: RP, BPP, RL, BPL. Error reduction for randomized algorithms. Relationships with nonuniformity and nondeterminism: BPP in P/Poly; BPP in Sigma_2 and Pi_2. Some randomized algorithms: Polynomial Identity Test in coRP, and Undirected ST Connectivity in RL. Arora-Barak Chapter 7. Goldreich Chapter 6. 2/28, 3/4 Interactive Proof. GNI in IP. #SAT in IP. IP = PSPACE. Arora-Barak Chapter 8. Goldreich Chapter 9.1. 3/6, 3/11 Introduction to Probabilistically Checkable Proof (PCP) and hardness of approximation. The statement of the PCP Theorem: NP = PCP(log n, 1). Equivalence between the PCP Theorem and the NP-hardness of Gap-3SAT_{1,s} for some constant s < 1. The statement of Hastad's theorem on 3-bit PCP and tight hardness result for MAX-3LIN, and tight hardness result for MAX-3SAT as a corollary. Arora-Barak Chapter 18. Goldreich Chapter 9.3. Lecture notes of this course and that course. 3/13 The FGLSS reduction and hardness of approximating MAX-CLIQUE. Lecture notes for Lecture 7 of this course . For a background on expanders, see this excellent survey article and lecture notes for this course. 3/25 Wrap up the discussion on the FGLSS reduction and polynomial hardness of MAX-CLIQUE. Digression: Explicit expanders and randomness-efficient reduction of both one-sided and two-sided errors. Linearity test. For FGLSS reduction, see lecture notes for Lecture 7 of this course . For a background on expanders, see this excellent survey article and lecture notes for this course. For linearity test, see lecture notes for Lecture 4 of this course . 3/27 Fourier analysis of Boolean functions and analysis of the BLR test. Local decoding of the Hadamad code. Get ready for NP in PCP[poly, 1]. Lecture notes for Lecture 4 of this course . Arora-Barak 18.4.1. 4/1 NP in PCP[n^2, 1]. 4/3, 4/8 Lower Bounds for Constant Depth Circuits. Alon-Spencer Ch. 11.2 - 11.3. 4/10, 4/15 Lower Bounds for Monotone Circuits. Section 4 of this excellent survey by Boppana and Sipser. 4/22, 4/24 Natural Proofs. The original paper by Razborov and Rudich. For background in crytography, see Foundations of Cryptography by Goldreich.