**Speaker:**
IBM/NYU/Columbia

**Location:**
Schapiro (CEPSR) Building, Columbia University 412

**Date:**
May 4, 2012, 9:30 a.m.

**Host:** Columbia University, New York

**Synopsis:**

The New York Area Theory Day is a semi-annual conference, aimed to bring together people in the New York Metropolitan area for one day of interaction and discussion. The Theory Day features several (usually 4-5) hour-long presentations by leading theoretical computer scientists about state-of-the-art advances in various areas. Some presentations give a survey of the latest advances in some area, while others may concentrate on a particular result. The meeting is free and open to everyone; in particular, students are encouraged to attend.

For directions, please go to: http://www.cs.columbia.edu/theory/directions.html

Program

9:30 - 10:00 Coffee and bagels

10:00 - 10:55 Prof. Michael Kearns (U Penn) Experiments in Social Computation

10:55 - 11:05 Short break

11:05 - 12:00 Prof. Eli Ben-Sasson (Technion and Microsoft Research New England) An Additive Combinatorics Approach Relating Rank to Communication Complexity

12:00 - 2:00 Lunch break

2:00 - 2:55 Dr. Aleksander Madry (Microsoft Research New England) Online Algorithms and the K-server Conjecture

2:55 - 3:15 Coffee break

3:15 - 4:10 Dr. Shubhangi Saraf (IAS) The Method of Multiplicities

To subscribe to our mailing list, follow instructions at http://www.cs.nyu.edu/mailman/listinfo/theory-ny

Organizers: Yevgeniy Dodis dodis@cs.nyu.edu Tal Rabin talr@us.ibm.com Baruch Schieber sbar@us.ibm.com Rocco Servedio rocco@cs.columbia.edu

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ABSTRACTS

Experiments in Social Computation

Michael Kearns University of Pennsylvania

What does the theory of computation have to say about the emerging phenomena of crowdsourcing and social computing? Most successful applications of crowdsourcing to date have been on problems we might consider "embarrassingly parallelizable" from a computational perspective. But the power of the social computation approach is already evident, and the road cleared for applying it to more challenging problems.

In part towards this goal, for a number of years we have been conducting controlled human-subject experiments in distributed social computation in networks with only limited and local communication. These experiments cast a number of traditional computational problems --- including graph coloring, consensus, independent set, market equilibria, and voting --- as games of strategic interaction in which subjects have financial incentives to collectively "compute" global solutions. I will overview and summarize the many behavioral findings from this line of experimentation, and draw broad comparisons to some of the predictions made by the theory of computation and microeconomics.

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An Additive Combinatorics Approach Relating Rank to Communication Complexity

Eli Ben-Sasson Technion and Microsoft Research New England

For a {0,1}-valued matrix M let CC(M) denote the deterministic communication complexity of the boolean function associated with M. It is well-known since the work of Mehlhorn and Schmidt [STOC 1982] that CC(M) is bounded from above by rank(M) and from below by log rank(M) where rank(M) denotes the rank of M over the field of real numbers. Determining where in this range lies the true worst-case value of CC(M) is a fundamental open problem in communication complexity. The state of the art is

log^{1.631} rank(M) < CC(M) < 0.415 rank(M),

the lower bound is by Kushilevitz [unpublished, 1995] and the upper bound is due to Kotlov [Journal of Graph Theory, 1996]. Lovasz and Saks [FOCS 1988] conjecture that CC(M) is closer to the lower bound, i.e., CC(M) < log^c(rank(M)) for some absolute constant c --- this is the famous ``log-rank conjecture'' --- but so far there has been no evidence to support it, even giving a slightly non-trivial (o(rank(M))) upper bound on the communication complexity.

Our main result is that, assuming the Polynomial Freiman-Ruzsa (PFR) conjecture in additive combinatorics, there exists a universal constant c such that

CC(M)< c rank(M)/ log rank(M).

Although our bound is stated using the rank of M over the reals, our proof goes by studying the problem over the finite field of size 2, and there we bring to bear a number of new tools from additive combinatorics which we hope will facilitate further progress on this perplexing question.

In more detail, our proof is based on the study of the ``approximate duality conjecture'' which was suggested by Ben-Sasson and Zewi [STOC 2011] and studied there in connection to the PFR conjecture. First we improve the bounds on approximate duality assuming the PFR conjecture. Then we use the approximate duality conjecture (with improved bounds) to get our upper bound on the communication complexity of low-rank martices.

Joint work with Shachar Lovett (IAS) and Noga Ron-Zewi (Technion)

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Online Algorithms and the K-server Conjecture

Aleksander Madry Microsoft Research New England

Traditionally, in the problems considered in optimization, one produces the solution only after the whole input is made available. However, in many real-world scenarios the input is revealed gradually, and one needs to make irrevocable decisions along the way while having only partial information on the whole input. This motivates us to develop models that allow us to address such scenarios.

In this talk, I will consider one of the most popular approaches to dealing with uncertainty in optimization: the online model and competitive analysis; and focus on a central problem in this area: the k-server problem. This problem captures many online scenarios - in particular, the widely studied caching problem - and is considered by many to be the "holy grail" problem of the field.

I will present a new randomized algorithm for the k-server problem that is the first online algorithm for this problem that achieves polylogarithmic competitiveness.

Based on joint work with Nikhil Bansal, Niv Buchbinder, and Joseph (Seffi) Naor.

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The Method of Multiplicities

Shubhangi Saraf IAS

Polynomials have played a fundamental role in the construction of objects with interesting combinatorial properties, such as error correcting codes, pseudorandom generators, randomness extractors etc. Somewhat strikingly, polynomials have also been found to be a powerful tool in the analysis of combinatorial parameters of objects that have some algebraic structure. This method of analysis has found applications in works on list-decoding of error correcting codes, constructions of randomness extractors, and in obtaining strong bounds for the size of Kakeya Sets. Remarkably, all these applications have relied on very simple and elementary properties of polynomials such as the sparsity of the zero sets of low degree polynomials.

In this talk we will discuss improvements on several of the results mentioned above by a more powerful application of polynomials that takes into account the information contained in the *derivatives* of the polynomials. We call this technique the ``method of multiplicities". The information about higher multiplicity vanishings of polynomials, which is encoded in the derivative polynomials, enables us to meaningfully reason about the zero sets of polynomials of degree much higher than the underlying field size.

We will discuss some of these applications of the method of multiplicities, to obtain improved constructions of error correcting codes, and qualitatively tighter analyses of Kakeya sets, and randomness extractors.

(Based on joint works with Zeev Dvir, Swastik Kopparty, Madhu Sudan, Sergey Yekhanin)

**Notes:**

Refreshments will be offered starting 15 minutes prior to the scheduled start of the talk.