CS 254 — Computational Complexity — Winter 2014

[general info]  [lecture notes] [homeworks] [midterm and project]

what's new

general information

Instructor: Luca Trevisan, Gates 474, Tel. 650 723-8879, email trevisan at stanford dot edu

TA: TongKe Xue email tkxue at stanford dot edu

Classes are Tuesdays and Thursdays, 2:15-3:30, Gates B12

Office hours:

References: the main reference for the course will be lecture notes. New lecture notes will be distributed after each lecture. A recommended textbook is

Another very good book, which covers only part of the topics of the course is About this course: Computational Complexity theory looks at the computational resources (time, memory, communication, ...) needed to solve computational problems that we care about, and it is especially concerned with the distinction between "tractable" problems, that we can solve with reasonable amount of resources, and "intractable" problems, that are beyond the power of existing, or conceivable, computers. It also looks at the trade-offs and relationships between different "modes" of computation (what if we use randomness, what if we are happy with approximate, rather than exact, solutions, what if we are happy with a program that works only for most possible inputs, rather than being universally correct, and so on). 

This course will roughly be divided into two parts: we will start with "basic" and "classical" material about time, space, P versus NP, polynomial hierarchy and so on, including  moderately modern and advanced material, such as the power of randomized algorithm, the complexity of counting problems, and the average-case complexity of problems. In the second part, we will focus on more research oriented material, to be chosen among: (i)  PCP and hardness of approximation; (ii) lower bounds for proofs and circuits; and (iii) derandomization and average-case complexity; (iv) quantum complexity theory.

There are at least two goals to this course. One is to demonstrate the surprising connections between computational problems that can be discovered by thinking abstractly about computations: this includes relations between learning theory and average-case complexity, the Nisan-Wigderson approach to turn intractability results into algorithms, the connection, exploited in PCP theory, between efficiency of proof-checking and complexity of approximation, and so on. The other goal is to use complexity theory as an "excuse" to learn about several tools of broad applicability in computer science such as expander graphs, discrete Fourier analysis, learning, and so on.

classes and lecture notes

    past classes

    1. Introduction. [pdf notes]
    2. Search and decision problems, hierarchy theorems. [pdf notes]
    3. Boolean circuits and NP-completeness, relativization. [pdf notes] (notes on relativizations will appear in the notes of Lecture 4)
    4. P versus NP relative to a random oracle, BPP, Adleman's theorem [pdf notes]
    5. Polynomial hierarchy, BPP in Sigma-2, Karp-Lipton. [pdf notes]
    6. Kannan's theorem, counting problems [pdf notes]
    7. Approximate counting, Valiant-Vazirani [pdf notes]
    8. Average-case complexity: definitions [pdf notes]
    9. Average-case complexity: NP-completeness [pdf notes]
    10. Pseudorandomness and derandomization [pdf notes]
    11. Parity is not in AC0: proof with polynomials [pdf notes]
    12. Parity is not in AC0: proof with the switching lemma [pdf notes]
    13. Natureal proofs. Notes in preparation, meanwhile you can check out these references: [Tim Gowers on Natural Proofs], [Luca on Natural Proofs], [Luca on Ryan's lower bound]
    14. Quantum computing: the model. Notes in preparation. Meanwhile you can read Section 1 (skipping 1.4) of [these notes]
    15. The PCP model. Notes in preparation, meanwhile (and for the next three lecture) a good reference is this paper of Radhakrishnan and Sudan
    16. Dinur's proof: amplification

    the tentative plan


  1. homework 1
  2. homework 2
  3. homework 3
  4. homework 4
  5. homework 5