Introduction to Computational Complexity (COMS 4326), Fall 2026, Toniann Pitassi

Introduction to Complexity Theory

COMS 4236, Spring 2026

Description

Stay Tuned.

General Information

Course Syllabus and Grading : Please read the following syllabus carefully. It includes information on grading, collaboration and use of AI policy, and important dates. Syllabus

Textbook and Reading Materials: We will post readings and lecture notes for each lecture. Supplementary recommended books are: Computational Complexity by Christos Papadimitriou; Computational Complexity A Modern Approach, by Arora and Barak.

Prerequisite: A very good background in CS Theory. You are expected to be familiar with the basic notions of Turing machines, decidability, nondeterminism, and NP-completeness, at the level of Chapters 3,4,7 of Sipser, Introduction to the Theory of Computation. You should also be comfortable with asymptotic notation (Big-O and Big-Omega), and basic run-time analysis of algorithm.

Grading: Stay tuned.

List of Topics :  Here is a preliminary list of core topics. For more information, go to the details of each lecture below.

  • Introduction to complexity theory; Turing machines; time complexity, the complexity classes P and NP, NP-completeness. Introduction to other notions of complexity that we will cover in the class and their relationships: space, randomness, communication complexity. Types of problems and solutions we will discuss: decision problems, optimization, search, approximation algorithms, randomized algorithms, worst-case versus average-case.

  • Time complexity. Complexity classes P, NP, PH, EXP. Complete problems.

  • Space complexity. Logspace, NL, RL, PSPACE. Graph connectivity, Branching Programs, Reversible and Catalytic Space

  • Circuit complexity: circuits and formulas. Lower bounds for constant-depth circuits.

  • Communication complexity.

  • Randomized algorithms, BPP and RP, Relationship between Hardness (worst-case, average-case time complexity) and Randomness.

  • Approximation algorithms, PCP theorem and consequences.

Materials

Lecture notes, homework assignments, and other materials will be posted here!

Lecture Notes and Slides: Here is the tentative plan for each lecture and corresponding readings. We will update these summaries as we progress through the semester.

Homework and Supplementary Material: