CS 221: Computational
Description | Topics |
| Grading | Readings | Related
Courses | Past Offerings
Instructor: Salil Vadhan (MD 337)
Shopping week office hours: Mon 1/25 2:30-5:30, Tue 1/26 10-12, Wed 1/27 2:30-5:30, Fri 1/29 10-11:30, 2:30-4.
TFs: Colin Jia Zheng (MD 138); Office Hour: Wed 6-7 PM in MD 2nd floor lounge, or by appointment
Assistant: Carol Harlow (MD 343, has handouts)
Room & time: MW 1-2:30, MD G-135
Sections: Tue 7-8 PM, MD 123
Course email address (use for all questions): firstname.lastname@example.org
Course website: http://www.courses.fas.harvard.edu/5812
Computational complexity aims to understand the fundamental limitations
and capabilities of efficient computation. For example, which
computational problems inherently require a huge running time to solve,
no matter how clever an algorithm one designs? This most basic question
of computational complexity is now understood to be both extremely difficult
and of great importance, as demonstrated by all the attention given to
the famous P vs. NP question. At the same time, however,
this is but one of many the fascinating issues addressed by complexity
theory (and covered in this course). First, running time will not
be the only computational resource we consider, but also space/memory,
nondeterminism, randomness, parallelism, communication, algebraic operations,
and quantum mechanics. We will also study a variety of types of computational
problems, such as decision, search, counting, optimization, and proof verification.
We will introduce an array of complexity classes to capture these resources
and problem types. We will use the powerful notions of reduction
and completeness to establish relationships between seemingly unrelated
problems, classes, and resources. Indeed, it is in discovering
such connections that complexity theory has had its greatest successes,
and we will see one of the most surprising ones: the
equivalence between probabilistic verification of mathematical proofs (PCPs)
and the complexity of finding approximate solutions to optimization problems.
We will also examine various approaches to separating P and NP, and more
generally to proving lower bounds on complexity. Finally, we will study
what happens when one relaxes the requirement for an algorithm to be "correct",
for example from worst-case complexity to average-case complexity or from
exact solutions to approximate solutions.
The material in this course can be of interest to a wide range of graduate
students and advanced undergraduates, ranging from those who plan to do
research in the theory of computation, to those working in other areas
of computer science and mathematics, to those interested in computational
aspects of other fields such as economics and physics.
Tentative List of Topics
Definite topics (Arora-Barak Chs 1-7, plus some of Chs 8,11)
Resources for computation (time, space, nondeterminism, randomness) and
their associated complexity classes.
Relationships among resources (P vs. NP and more)
Reductions & completeness
- Provably intractable problems: hierarchy thms, EXPSPACE-completeness
- Space complexity: PSPACE, L, NL
- Randomized computation: RP, BPP
- Alternation: the polynomial hierarchy (PH), time-space tradeoffs for SAT
- Relativization (why diagonalization can't resolve P vs NP)
- Basic circuit complexity (P/poly, NC)
Interactive proofs (AM, MA, IP)
Probabilistically checkable proofs (PCP) and nonapproximability
Possible topics (material from Arora-Barak Chs 8,13,14,16,17,19):
- Proofs of IP=PSPACE, PCP Thm(s)
- Unique Games Conjecture
- Parity not in AC^0
- Average-case complexity
- Counting: #P, Toda's Thm, approximate counting
- Communication complexity and applications
Algebraic complexity: VNP, VP, Permanent vs. Determinant
Quantum computation: BQP, Shor's Factoring algorithm
The prerequisite for this course is Computer Science 121 (or the equivalent
at another university) with a good grade (B+ or higher). If
you do not formally meet this requirement but still wish to take this course,
you must (a) come to my office hours during shopping week to discuss your
preparation for the course, (b) do Problem Set 0, and (c) come to my office
hours again before add/drop date to discuss your performance on PS0 and comfort
in the course so far. For students meeting the prerequisite,
Problem Set 0 is optional, but we encourage you to work through the problems.
most important topics that we will be assuming from CS 121 is comfort with
Turing machines, computability, asymptotic running time, and NP-completeness.
General mathematical maturity, e.g. comfort with proofs, basic discrete
probability, & combinatorics, will also be assumed.
Grading & Problem Sets
The requirements of the course:
The biweekly problem sets will typically be handed out on Wednesdays, and due two weeks later on Thursday by 5 PM. You
are allowed 12 late days for the semester, of which at most 5 can be used
on any individual problem set. (1 late day = 24 hours exactly). For
an exception, you must have your senior tutor (for undergrads) or your
advisor (for graduate students) contact me.
Biweekly problem sets. Your solutions must be typed and submitted
- Typing up scribe notes for a few lectures during the semester.
Take-home final exam. (Small chance that this will be switched this to an essay+presentation.)
- Class participation.
The problem sets will be challenging, so be sure to start them early.
You are encouraged to discuss the course material and the homework problems
with each other in small groups (2-3 people), as long as you list all discussion
partners on your problem set. Discussion of homework problems may include
brainstorming and verbally walking through possible solutions, but should
not include one person telling the others how to solve the problem.
In addition, each person must write up their solutions independently.
You may not look at another student's written solutions.
The required text is the new book, Computational Complexity: A Modern Approach, by Sanjeev Arora and Boaz Barak. It should be available at the Coop, and be on reserve at McKay library. You can also get it from:
Since the book is new, there are likely to be some typos and errors. If you find any, please email us, so we can notify the class and the authors.
This course is generally offered every other year. Information from previous times I taught it, including Q evaluations, is available at http://seas.harvard.edu/~salil/cs221