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.

  • There will be 5-6 presentations given by students. Each presentation will involve 3 people: the main presenter, the supporting presenter (helper) and the reviewer. Here is the list of presentation topics to choose from, with details on the selection process. Each student should select one topic where they are the main presenter, one topic where they are the supporting presenter, and one topic where they are the reviewer. You should have received an email on Feb 18 with link to a google docs where you can sign up for your topic. At the bottom of the google docs sheet is an optional sign up to meet with me this week if you are undecided or want to discuss your options. Presentation topics

  • Main Presenter: 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.

  • Supporter (helper): read the paper carefully, and meet with main presenter prior to their lecture to understand the paper(s) and to go over the presentation and give feedback (generally, the presenters should share their preliminary slides or lecture notes with the supporters for feedback a week before the presentation). Supporters should be available to meet with the main presenter to work on the presentation and to help come up with questions/answers during the lecture.

  • Reviewer: responsible for reading the paper carefully and presenting a critical review. This can include pros and cons of the paper (was it well written and well presented; did it solve an important problem), directions and problems left open by the paper, and stimulate the audience by posing questions.

  • Meeting with instructor: the team will 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 me).

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. Lecture 1
  • Jan 28: Deterministic Communication, Combinatorial characterization and LogRank Conjecture, Intro Randomized Communication. Lecture 2
  • Feb 4 : Randomized Communication: Examples; Randomized models; error reduction; compression of randomness; Public Coin vs Private Coin Lecture 3
  • Feb 11 : Nondeterministic Communication, Randomized Communication (Equivalence with Distributional complexity). Lecture 4
  • Feb 18 : Lower Bounds in Communication Complexity via Discrepancy Lecture 5
  • Feb 25 : NOF Communication Complexity: Intro to NOF, cylinder intersections, applications in additive combinatorics Lecture 6
  • Mar 4 : Lifting I, Lifting decision tree lower bounds to CC lower bounds and Applications
  • Mar 11 : Lifting II: Randomized Lifting and Applications.
  • Mar 18: No Class (Spring Break Week)
  • Mar 25: Applications cont'd
  • Apr 1: Presentations: Game Theory, Property Testing
  • Apr 8: No Class
  • Apr 15: Presentations: Information Complexity, XOR Lemma
  • Apr 21: Presentations: Matrix Multiplication, Quantum
  • Apr 29 (Last Class): Wrapup, Open Problems

Homeworks: 

  • Coming Soon.