CS 254 — Computational Complexity — Winter 2012
[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 Mondays-Wednesdays, 4;15-5:30pm in 102 Hewlett
Office hours:
- Luca: Thursdays, 2:30-3:30, or by appointment, Gates 474
- TongKe: Mondays and Wednesdays, 6:30-8:30pm, Gates B24A
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.
past lectures
- 01/09. Introduction.
- 01/11. P vs NP, deterministic hierarchy theorem.
- Notes [pdf]
- Arora-Barak 2.1, 2.5, 3.1
- 01/18. Boolean circuits, BPP, error-reduction for randomized algorithms, Adleman's theorem.
- Notes [pdf]
- Arora-Barak 6.1, 6.5, 7.1, 7.2, 7.3, 7.4,7.5
- 01/23. Polynomial hierarchy, BPP in Sigma2, Karp-Lipton
- Notes [pdf]
- Arora-Barak 7.5, 5.1, 5.2
- 01/25. Kannan's theorem, #P
- 01/30. Approximate counting
- 02/01. Valiant-Vazirani theorem and more on approximate counting and approximate sampling
- Notes in preparation. See Chapter 5 of this online book
as a reference for the material on approximate counting and sampling
- 02/06. Average-case complexity
- 02/08. PCP Theorem
- 02/13 Inapproximability
- 02/15 More inapproximability
- 02/22 Parity not in AC0
- 02/27 More on parity not in AC0
- <02/29 Pseudorandomness and derandomization
- These notes cover the material
of this lecture and of next lecture
future lectures
- The Nisan-Wigderson pseudorandom generator
- Natural proofs
homeworks
- Problem set 1 out Jan 11, due Jan 20
- Problem set 2 out Jan 18, due Jan 27
- Problem set 3 out Jan 25, due Feb 3
- Problem set 4 out Feb 1, due Feb 10
- Problem set 5 out Feb 15, due Feb 24
- Problem set 6 out Jan 23, due Mar 2
exams