### DIMACS Mixer Series - Mixer II

#### Abstracts:


1.

Universal Exploration Sequences
Michael Koucky, Rutgers University

An s-t-connectivity question in an undirected graph can be answered
using a very simple randomized algorithm. This randomized algorithm gives
a rise to an interesting combinatorial object - universal traversal
sequences, that were defined by Aleliunas et al. Siblings of universal
traversal sequences are universal exploration sequences.

In the talk we will give a definition of universal exploration
sequences, mention their basic properties, and show some explicit
constructions.

2.

Communication complexity and Secure Function Evaluation
Kobbi Nissim

A secure function evaluation protocol allows two parties to jointly compute a
function f(x,y) of their inputs in a manner not leaking more information than
necessary. A major result in this field is: "any function f that can be
computed using polynomial resources can be computed securely using polynomial
resources" (where resources' refers to communication and computation). This
result follows by a general transformation from any Boolean circuit for f to a
secure protocol that evaluates f.

We suggest new methodologies for the design of efficient secure protocols that
differ with respect to their underlying computational models. One methodology
utilizes the communication complexity tree (or branching program)
representation of f. We start with an efficient (insecure) protocol for f and
transform it into a secure protocol in a communication preserving manner. The
second methodology uses a circuit computing f, enhanced with look-up tables. It
is possible to simulate any RAM machine in this model with polylogarithmic
blowup. Hence it is possible to start with a computation of $f$ on a RAM
machine and transform it into a secure protocol.

There are many applications of these new methodologies resulting n protocols
efficient either in communication or in computation. In particular, we
exemplify a protocol for the "millionaires problem", where two participants
want to compare their values but reveal no other information. Our protocol is
more efficient than previously known ones in either communication or
computation.

The talk will focus on the computational models underlying our results, and
on some of the open questions.

Joint work with Moni Naor.

3.

Speaker:  Carl Pomerance, Bell Labs - Lucent Technologies

Title:  Carmichael's Function

Abstract:  Carmichael's function of $n$ is the order
of the largest cyclic subgroup of the multiplicative
group of integers modulo $n$.  What could be simpler?
Actually, a lot could!  Related to factoring, primality
testing, Artin's conjecture, and Euler's function,
there is still much that we don't know, and in some
of the problems it is even hard to find the right conjecture.

4.

Venkatesh Srinivasan, DIMACS/IAS
On the advantage over a random assignment

We initiate the study of a new measure of approximation. This measure
compares the performance of an approximation algorithm to the
"random assignment" algorithm. Since the random assignment algorithm
is known to give the best approximation algorithm for many optimization
problems, we believe that this measure is fundamentally interesting.

We study this measure for the optimization problem, Max-Lin-2, in
which we need to maximize the number of satisfied linear equations
in a system of linear equations modulo 2. The main ideas we use in
our results are from Fourier analysis and derandomization. We will also
mention some algorithmic applications of our results for Max-Lin-2.

This is joint work with Johan Hastad.

5.

Mario Szegedy, Rutgers University
Computing Boolean Functions from Multiple
Faulty Copies of Input Bits

Abstract.

Suppose, we want to compute a Boolean function f, but instead
of receiving the input, we only get some epsilon-faulty copies
of each input bit. A typical solution in this case is to take the
majority value of the faulty bits for each individual input bit and
apply f on the majority values. We show that if f:{0,1}^{n} -> {0,1}
and epsilon are known, the best construction function, F, is often
not the trivial. In particular, in many cases the best F cannot be
written as a composition of some functions with f, and in addition
it is better to use a randomized F than a deterministic one. Our
Decision tree complexity, introduced by Reischuk at. al. We
fully characterize this complexity measure using a lemma
that gives a necessary and sufficient condition for the value
of a Boolean function to remain constant under independent
random changes of its input bits. This lemma may be interesting
on its own right.

6.
TITLE:
Informational Complexity and the Direct Sum Problem
for Simultaneous Message Complexity

AUTHORS:
Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, Andrew Yao

SPEAKER:
Anthony Wirth

ABSTRACT:
Given $m$ copies of the same problem, does it take $m$ times
the amount of resources to solve these $m$ problems? This is the direct
sum problem, a fundamental question that has been studied in many
computational models. We study this question in the simultaneous message
(SM) model of communication introduced by Yao.

The equality problem for $n$-bit strings is well known to have SM
complexity $\Theta(\sqrt n)$. We prove that solving $m$ copies of the
problem has complexity $\Omega(m\sqrt n)$; the best lower bound provable
using previously known techniques is $\Omega(\sqrt{mn})$. We also prove
similar lower bounds on certain Boolean combinations of multiple copies
of the equality function. These results can be generalized to a broader
class of functions.

We introduce a new notion of {\em informational complexity} which is
related to SM complexity and has nice direct sum properties. This notion
is used as a tool to prove the above results; it appears to be quite
powerful and may be of independent interest.

KEYWORDS:
Communication Complexity, Direct Sum Problem,
Simultaneous Message Complexity, Probabilistic Method

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