Communication Complexity and Applications (COMS 699), Spring 2026, Toniann Pitassi

Topics in Communication Complexity

COMS 6998, Spring 2026

Description

One of the most basic challenges in the theory of computation is proving computational limitations for well-studied important problems in a variety of models. Communication complexity has played a central role in understanding information bottlenecks for many computational models as well as to many seemingly unrelated problems. For example, communication complexity has been the main tool used to establish state-of-the-art lower bounds in the following areas: graph theory, circuit complexity, game theory, proof complexity, property testing, extension complexity, distributed computing and learning theory. In this course we will cover the foundations of communication complexity, and applications to a wide variety of other areas. In this course we will survey the new and burgeoning field of communication complexity, exploring the connections between a variety of other areas. There will be an emphasis on open problems and research techniques. This course is an advanced graduate level seminar, where most lectures will be given by students taking the class (with prior feedback and support from the instructors and fellow students). The specific papers covered will be selected by the instructors, biased by their interests, and (especially for the later lectures) also by the interests of the students taking the class.

General Information

Textbook and Reading Materials: We will post readings and lecture notes for each lecture. Supplementary recommended books are: Communication Complexity by Kushilevitz and Nisan, and Communication Complexity and Applications by Rao and Yehudayoff.

Prerequisite: An undergraduate (3rd or 4th year) course in Complexity Theory or Theory of Computation and a solid background in Linear Algebra.

Grading: There will be one or two homeworks based on the material taught in the first 8 lectures which will comprise 20 percent of the grade. In the second half of the course, most lectures will be delivered by students. There will be 1 or 2 students in charge of the lecture (presenting the assigned paper to the class), and 1 or 2 students supporting the presenter(s). Every student is expected to take the role of a presenter 1-2 times and the role of a supporter 1-2 times throughout the semester. This will comprise 60 percent of the grade. The final 20 percent will be based on attendance and active participation during all lectures and presentations.

Presentation Details: Here is the summary of what is required for the presentations.

  • Presenter(s): have to read the paper carefully, think about the right way to teach it to the audience, how to structure their presentation, what to include and not include in the given time (some papers may be quite long), what the main message is, and determine the mechanism they prefer (slides or blackboard). They should also answer student questions during the lecture.

  • Supporters(s): have to read the paper carefully, and meet with presenters prior to their lecture to go over the presentation and give feedback (generally, the presenters should share their preliminary slides or lecture notes with the supporters for feedback at least a day before the class they are teaching). Supporters should be generally available over email to answer presenters' questions prior to the lecutre, and prepared to help answer stduents' questions during the lecture if needed.

  • Meeting with instructors: the team will need to meet with the instructor prior to their presentation for feedback.

  • Materials: the team should also provide materials for rest of the class, in the form of notes or slides summarizing the lecture (again with feedback from the teaching staff).

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.

  • Jan 21: Intro to communication complexity, Models of communication complexity, Important examples. Notes: Lecture 1
  • Jan 28: Deterministic Communication, Combinatorial characterization, Lower Bounds.
  • Feb 4 : Deterministic Communication, Logrank Conjecture.
  • Feb 11 : Randomized Communication, Discrepancy Lower Bounds.
  • Feb 18 : Lower Bounds in Communication Complexity via Lifting, I.
  • Feb 25 : Lifting, II.
  • Mar 4 : Applications: Circuit Complexity, Extension Complexity.
  • Mar 11 : NOF Communication Complexity.
  • Mar 18: No Lecture (Spring Break Week)
  • Mar 25: Applications cont'd
  • Apr 1: No Lecture (Presentation Preparation)
  • Apr 8: No Lecture (Presentation Preparation)
  • Apr 15:
  • Apr 21:
  • Apr 29: Last Class

Homeworks: 

  • Coming Soon.

Supplemental Material: