### New York Area Theory Day - Fall 2017 Friday, December 1, 2017 Organized by: IBM/NYU/Columbia Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, Auditorium 109, New York. (Directions) Program


9:30 - 10:00   Coffee and bagels

10:00 - 10:55   Prof. Christos Papadimitriou
A Computer Scientist Thinks about the Brain

10:55 - 11:15   Coffee break

11:15 - 12:05   Short talks:
Shay Solomon, Ilya Razenshteyn, Sepideh Mahabadi

12:05 -  2:00   Lunch break

2:00 -  2:55   Dr. Ilan Komargodski
White-Box vs. Black-Box Complexity of Search Problems:
Ramsey and Graph Property Testing

2:55 -  3:15   Short talk: Sandro Coretti

3:15 -  3:35   Coffee break

3:35 -  3:55   Short talk: Greg Bodwin

3:55 -  4:50   Prof. Virginia Vassilevska Williams
Fine-Grained Complexity of Problems in P

5:00 -  7:00   Follow-up social

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Organizers:

Alex Andoni
Yevgeniy Dodis
Krzysztof Onak

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LONG TALKS
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Title: A Computer Scientist Thinks about the Brain

Abstract: Understanding the ways in which the Brain gives rise to the Mind
(memory, behavior, cognition, intelligence, language) is the most challenging
problem facing science today. As the answer seems likely to be at least partly
computational, computer scientists should work on this problem. Using a
simplified model of Hebbian plasticity and employing random graph theory and
Bernoulli approximations we reproduce a form of association between memories
recently observed experimentally in humans. We discuss several implications and
extensions. (Joint work with W. Maass and S. Vempala).

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Ilan Komargodski (Cornell Tech)

Title: White-Box vs. Black-Box Complexity of Search Problems: Ramsey and Graph
Property Testing

Abstract: Ramsey theory assures us that in any graph there is a clique or
independent set of a certain size, roughly logarithmic in the graph size. But
how difficult is it to find the clique or independent set? This problem is in
TFNP, the class of search problems with guaranteed solutions.  If the graph is
given explicitly, then it is possible to do so while examining a linear number
of edges. If the graph is given by a black-box, where to figure out whether a
certain edge exists the box should be queried, then a large number of queries
must be issued.

1) What if one is given a program or circuit ("white-box") for computing the
existence of an edge. Does the search problem remain hard?
2) Can we generically translate all TFNP black-box hardness into white-box
hardness?
3) Does the problem remain hard if the black-box instance is small?

We will answer all of these questions and discuss related questions in the
setting of property testing.

Joint work with Moni Naor and Eylon Yogev.

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Virginia Vassilevska Williams (MIT)
Title: Fine-Grained Complexity of Problems in P

Abstract: A central goal of algorithmic research is to determine how fast
computational problems can be solved in the worst case. Theorems from complexity
theory state that there are problems that, on inputs of size n, can be solved in
t(n) time but not in t(n)^{1-eps} time for eps>0. The main challenge is to
determine where in this hierarchy various natural and important problems lie.
Throughout the years, many ingenious algorithmic techniques have been developed
and applied to obtain blazingly fast algorithms for many problems. Nevertheless,
for many other central problems, the best known running times are essentially
those of their classical algorithms from the 1950s and 1960s.

Unconditional lower bounds seem very difficult to obtain, and so practically all
known time lower bounds are conditional. For years, the main tool for proving
hardness of computational problems have been NP-hardness reductions, basing
hardness on P\neq NP. However, when we care about the exact running time (as
opposed to merely polynomial vs non-polynomial), NP-hardness is not applicable,
especially if the problem is already solvable in polynomial time. In recent
years, a new theory has been developed, based on fine-grained reductions''
that focus on exact running times. Mimicking NP-hardness, the approach is to (1)
select a key problem X that is conjectured to require essentially t(n) time for
some t, and (2) reduce X in a fine-grained way to many important problems. This
approach has led to the discovery of many meaningful relationships between
problems, and even sometimes to equivalence classes.

The main key problems used to base hardness on have been: the 3SUM problem, the
CNF-SAT problem (based on the Strong Exponential TIme Hypothesis (SETH)) and the
All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has
flourished in particular in recent years showing that the classical algorithms
are optimal for problems such as Approximate Diameter, Edit Distance, Frechet
Distance, Longest Common Subsequence etc.

In this talk I will give an overview of the current progress in this area of
study, and will highlight some exciting new developments.

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SHORT TALKS
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Shay Solomon (IBM Research)
Title: Fully Dynamic Maximal Matching

Abstract: Graph matching is one of the most well-studied problems in
combinatorial optimization, with a reach plethora of applications. Nevertheless,
networks, which aim to model the constantly changing physical world.

Focusing on the maximal matching problem, there is a trivial linear time
algorithm for computing such a matching in static networks, which naturally
translates into an algorithm with a constant amortized cost in the incremental
case. In this talk I will discuss some of the challenges that we overcame to
obtain a constant amortized cost algorithm in the fully dynamic case.

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Ilya Razenshteyn (Columbia)
Title: Practical Data-Dependent Metric Compression with Provable Guarantees

Abstract: How well can one compress a dataset of points from a high-dimensional
space while preserving pairwise distances? Indyk and Wagner have recently
obtained almost optimal bounds for this problem, but their construction (based
on hierarchical clustering) is not practical. In this talk, I will show a new
practical, quadtree-based compression scheme, whose provable performance
essentially matches that of the result of Indyk and Wagner.

In addition to the theoretical results, we will see experimental comparison of
the new scheme and Product Quantization (PQ)--one of the most popular heuristics
for distance-preserving compression--on several datasets. Unlike PQ and other
heuristics that rely on the clusterability of the dataset, the new algorithm
ends up being more robust.

The talk is based on a joint work with Piotr Indyk and Tal Wagner.

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Title: Set Cover in Sub-linear Time

Abstract: We study the classic set cover problem from the perspective of
sub-linear algorithms. Given access to a collection of $m$ sets over $n$
elements in the query model, we show that sub-linear algorithms derived from
existing techniques have almost tight query complexities.

On one hand, first we show that an adaptation of streaming algorithms to the
sub-linear query model returns an $\alpha$-approximate cover using $\tilde O(m(n/k)^{1/(\alpha-1)} + nk)$ queries to the input, where $k$ denotes the value
of a minimum set cover. We then complement this upper bound by proving that for
lower values of $k$, the required number of queries is $\tilde Omega(m(n/k)^{1/(2\alpha)})$, even for estimating the optimal cover size.
Moreover, we prove that even checking whether a given collection of sets covers
all the elements would require $\Omega(nk)$ queries. These two lower bounds
provide strong evidence that the upper bound is almost tight for certain values
of the parameter $k$. On the other hand, we show that this bound is not optimal
for larger values of the parameter $k$, as there exists a
$(1+\eps)$-approximation algorithm with $\tilde O(mn/k\eps^2)$ queries. We show
that this bound is essentially tight for sufficiently small constant $\eps$, by
establishing a lower bound of $\tilde Omega(mn/k)$ query complexity.

This is a joint work with Piotr Indyk, Ronitt Rubinfeld, Ali Vakilian, and Anak
Yodpinyanee.

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Sandro Coretti (New York University)
Title: Random Oracles and Non-Uniformity

Abstract: Hash functions are ubiquitous in cryptography. They are widely used in
practice to build one-way functions (OWFs), collision-resistant hash functions
(CRHFs), pseudorandom functions/generators (PRFs/PRGs), message authentication
codes (MACs), etc. Moreover, they are often used together with other
computational assumptions to prove the security of higher-level applications,
such as Fiat-Shamir heuristics for signature schemes, full-domain-hash
signatures, or OAEP encryption, among many others.

The security of such schemes is commonly analyzed in the random-oracle model
(ROM), in which the hash function is modeled as a truly random function that can
be queried by an attacker at a bounded number of points. The traditional ROM,
however, does not account for the fact that a non-uniform attacker may obtain
arbitrary advice about the hash function H before attacking a construction. As a
result, bounds derived in the ROM do not accurately reflect security against
such attackers.

To remedy this situation, Unruh (CRYPTO ’07) considered the auxiliary-input ROM
(AI-ROM), in which an attacker may obtain (a bounded amount of) arbitrary
leakage about the random oracle before attacking a cryptographic scheme. In this
talk we discuss significant improvements to Unruh's presampling technique,
which, intuitively, allows a leaky random oracle to be replaced by one that is
adversarially fixed on a subset of the coordinates but uniformly random
otherwise. We also show how this improved presampling leads to better and easily
derivable AI-ROM security bounds for a number of cryptographic applications. We
conclude by noting that in several cases (such as OWFs, PRGs, and CRHFs) there
are still significant gaps between the derived bounds and the best attacks,
leaving interesting open problems.

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Greg Bodwin (MIT)
Title: Compressing Graphs While Preserving Their Distances

Abstract: TBA