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{\bf COMS 6998 Communication Complexity and Applications, Spring 2022 } \\
{\bf Time/Location: Wednesdays 2:10-4 }  \\
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\noindent{\bf Instructor: Toniann Pitassi } \\
%\noindent{Office: 2305A Sandford Fleming Building} \\
\noindent{email: toni@cs.columbia.edu} \\
\noindent{Office hours: by appointment} \\

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\noindent {\bf Website:} 
${\rm www.cs.columbia.edu/}\sim{\rm toni/Courses/CC2022/CC2022.html}$
\\
Refer to this site periodically for important announcements and handouts,
including lecture notes and paper links. 

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\noindent {\bf Course 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.
Prerequisites are a solid background in
linear algebra, and an undergraduate course in complexity theory
or theory of computing.

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\noindent {\bf Tentative Course Outline.}
\begin{itemize}

\item[(I)] Introduction to two-player communication complexity.
Basic
concepts and definitions; motivation;
important cc problems (equality, set disjointness, inner product);
a brief introduction to the vast applications of 
communication complexity.

\item[(II)] Flavors of communication 
\begin{enumerate}
\item Deterministic 
\item Randomized (Distributional Complexity and Yao's theorem,
Public versus Private coins, Discrepancy method
\item Nondeterministic 
communication complexity 
\item Relationships between the models.
\end{enumerate}

\item[(III)] Lower bounds and lower bound methods. 
\begin{enumerate}
\item Fooling sets 
\item rank and logrank conjecture
\item discrepancy method 
\item corruption (set disjointness lower bound)
\item lifting (query complexity and polynomial degree to communication
complexity, strong link to TFNP classes and proof complexity.)
\item information complexity
\item important matrix norms and duality.
\end{enumerate}

\item[(IV)] Applications (We will cover the first 4 topics and
selected additional ones based on your preferences.)
\begin{enumerate}
\item Lifting query complexity to circuit complexity (KW games)
\item Lifting query complexity to extended formulations
\item Learning theory (via lifting)
\item Graph theory (Clique versus independent set)
\item Branching programs
\item Data structures
\item Streaming
\item Differential privacy
\item Mechanism Design
\item Property Testing
\end{enumerate}

\item[(V)] (Time Permitting) Number-on-forehead model of multiparty communication complexity.
Randomized, nondeterministic complexity in this model.
Lower bounds via the BNS discrepancy method and generalized
discrepancy.
Applications including:
Proof complexity lower bounds via NOF communication complexity,
ACC circuit lower bounds,
connections with additive combinatorics.


\end{itemize}



\noindent{\bf Grading and Assignments.}
Grading will be based on 2-3 assignments which will
be handed out during the semester, plus a short presentation.
More details on the presentation will be given in the first
few weeks of the class. You may work alone or in
small groups, and I will give a list of suggested topics/papers.

%The work you submit must be your own.
%You may discuss problems with each other;
%however you should prepare written solutions alone.
Class attendence is manditory and you are encouraged to
ask questions in class.
I will present many open problems during the course and
hope that some of you will solve some of these problems!
It is a great area with lots of connections to other problems,
and a wealth of interesting open problems.

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\noindent {\bf Course Materials.} Each lecture will include lecture notes and supplemental reading material, 
available on the website. Supplementary recommended books include: {\it Communication Complexity} by Kushilevitz and Nisan, and {\it Communication Complexity and Applications} by Rao and Yehudayoff.
The course webpage will have links to all lecture notes
and additional materials (such as links to papers and lecture notes 
from similar courses.)

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