Computational Complexity

16:198:538              Fall 2019


Instructor: Shubhangi Saraf


Timing:  Wednesday 1:40 pm – 4:40 pm

Location:  1:40 pm – 3 pm TIL 257; 3:20 pm – 4:40 pm TIL 258.     (Livingston Campus)

Office hours: Mon 3pm – 4 pm (Hill 426)


Description: This course will serve as a graduate course in complexity theory.


Computational complexity is the study of what computational tasks can be achieved efficiently, or with limited computational resources. Though computer scientists and mathematicians have intensively studied this topic in the last several decades, we still understand very little about computational efficiency.  Although there have been some remarkable results giving some answers and insights into some of the questions, most of major questions are still unanswered. Indeed it seems almost shocking that we still don't know the answers to them. In this course we will cover classical results as well as more recent results leading up to the state of the art in the field. Along the way we will encounter many surprising connections, beautiful mathematics, and a host of intriguing questions.


Recommended Text: "Computational Complexity: A Modern Approach" by Arora and Barak.

Also available online:


Another great book (also available online):

"Computational Complexity: A conceptual Perspective" by Oded Goldreich


Prerequisites: It will be helpful to have some background in algorithms/discrete math, but no formal prerequisite will be enforced. If you do not satisfy the official prerequisites but are still interested in registering for the course, and you are concerned if you will be sufficiently prepared for the course, send an email to

The only real prerequisite is some mathematical maturity.


Homework/grading: There will be 3 problem sets. There will also be a final project.


Hw 1 – Due October 2 (in class)

Hw 2 – Due October 30 (in class)

Hw-3 – Due December 11 (in class)




§  Lecture 1 (09/04): Administrative details, course overview, Turing machines, complexity classes.   

§  Lecture 2 (09/11): P, NP, Cook-Levin Theorem   

§  Lecture 3 (09/18): Hierarchy theorems, Ladner’s Theorem

§  Lecture 4 (09/25): Oracles and relativization, Space bounded complexity

§  Lecture 5 (10/02): More space bounded complexity

§  Lecture 6 (10/09): NL, co-NL, polynomial hierarchy

§  Lecture 7 (10/16): Polynomial Hierarchy and alternating Turing machines

§  Lecture 8 (10/23): Randomized computation

§  Lecture 9 (10/30): More randomized computatios

§  Lecture 10 (11/6): Interactive proofs, AM, MA

§  Lecture 11 (11/13): Goldwasser-Sipser set lower bound protocol

§  Lecture 12 (11/20): IP = PSPACE

§                      (11/27): NO CLASS. (Wednesday is Friday schedule, THANKSGIVING BREAK)

§  Lecture 13 (12/04): PCP Theorem

§  Lecture 14 (12/11): PCP Theorem II









Tentative (and partial) list of topics:

·      Hierarchy theorems, Diagonalization, Relativization,

·      Alternations, NP completeness,

·      Space bounded computation, sublinear space algorithms

·      Randomness in computation, BPP, RP

·      Interactive proofs

·      Circuit lower bounds

·      Hardness versus randomness – derandomization, pseudorandom generators

·      PCP theorem, hardness of approximation