Program
10:00 - 10:55 Prof. Michael Kearns (U Penn)
Experiments in Social Computation
11:05 - 12:00 Prof. Eli Ben-Sasson (Technion + MSR-NE)
An Additive Combinatorics Approach to the Log-rank Conjecture
2:00 - 2:55 Dr. Aleksander Madry (Microsoft Research New England)
Online Algorithms and the K-server Conjecture
3:15 - 4:10 Dr. Shubhangi Saraf (IAS)
The Method of Multiplicities
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 to the Log-rank Conjecture
Eli Ben-Sasson
Technion and MSR-NE
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the lower bound is by Kushilevitz [unpublished, 1995] and the upper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,
and Sergey Yekhanin)