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:
- Luca: Tuesdays, 4-5:30pm, or by appointment, Gates 474
- TongKe: Wednesdays, 3-5pm, Gates 5th floor lounge
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 classes
- Introduction. [pdf notes]
- Search and decision problems, hierarchy theorems. [pdf notes]
- Boolean circuits and NP-completeness, relativization. [pdf notes] (notes on relativizations will appear in the notes of Lecture 4)
- P versus NP relative to a random oracle, BPP, Adleman's theorem [pdf notes]
- Polynomial hierarchy, BPP in Sigma-2, Karp-Lipton. [pdf notes]
- Kannan's theorem, counting problems [pdf notes]
- Approximate counting, Valiant-Vazirani [pdf notes]
- Average-case complexity: definitions [pdf notes]
- Average-case complexity: NP-completeness [pdf notes]
- Pseudorandomness and derandomization [pdf notes]
- Parity is not in AC0: proof with polynomials [pdf notes]
- Parity is not in AC0: proof with the switching lemma [pdf notes]
- 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]
- Quantum computing: the model. Notes in preparation. Meanwhile you can read Section 1 (skipping 1.4) of [these notes]
- The PCP model. Notes in preparation, meanwhile (and for the next three lecture) a good reference is this paper of Radhakrishnan and Sudan
- Dinur's proof: amplification
the tentative plan
- Lecture 1: Introduction, P vs NP problem, deterministic
hierarchy theorem
- Lecture 2: Relativization, boolean circuits
- Lecture 3: BPP, Adleman's theorem
- Lecture 4: Polynomial hierarchy, BPP in Sigma2
- Lecture 5: Karp-Lipton and Valiant-Vazirani
- Lecture 6: Approximate counting
- Lecture 7: Average-case complexity
- Lecture 8: Pseudorandomness
- Lecture 9: Natural Proofs
- Lecture 10: Parity not in AC0 (Razborov-Smolensky)
- Lecture 11: Parity no in AC0 (Switching lemma)
- Lecture 12: Quantum complexity theory: the model and Grover search
- Lecture 13: Quantum complexity theory: search lower bounds
- Lecture 14: PCP theorem: statement and applications
- Lecture 15: Dinur's proof: amplification
- Lecture 16: Dinur's proof: range reduction
- Lecture 17: Applications to inapproximability
- Lecture 18: Unique games
homeworks
- homework 1
- homework 2
- homework 3
- homework 4
- homework 5
exams