CS 221: Computational Complexity
Spring 2010


Description | Topics | Prerequisites | 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): cs221@seas.harvard.edu
Course website: http://www.courses.fas.harvard.edu/5812

Course Description

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)

Possible topics (material from Arora-Barak Chs 8,13,14,16,17,19):


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.

The 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.

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.

Related Courses

Past Offerings

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