Pacific Institute for the Mathematical Sciences - PIMS

Two needles in exponential haystacks

Erdös Magic, aka The Probabilistic Method, is a powerful tool for
proving the existence of a combinatorial object, such as a coloring.
A probability space is created for which the probability of success
is positive. Hence the desired object must exist. But where is it?
Here we examine instances in which the probability is exponentially
small so that a randomized algorithm would not be in P. Nonetheless,
we give two recent startling successes.

Bansal: A quarter century ago this speaker showed that given n sets
on n vertices there is a two-coloring so that all discrepancies are
O(n√). He long conjectured that no polynomial time algorithm
could find the coloring. Wrong! Nikhil Bansal, making ingenious use of
semidefinite programming, finds the coloring and much more.

Moser: Even longer ago, László Lovász, with the Lovász Local
Lemma, showed (roughly!) that when bad events are mostly independent
there is a positive probability that the random object has no bad events. Robin Moser
gives a simple "fix-it" randomized algorithm to find the object. The proof that the algorithm
works, however, is most original. It gives a new and seemingly quite different proof
of the Local Lemma itself.

When the probabilistic method sieves an event with exponentially
small probability the usual randomized algorithms will not find an actualization.
We discuss two recent startling successes: Moser et.al. on the Lovász Local
Lemma and Bansal on the speaker's "Six Standard Deviations Suffice."

Location: Mechanical Engineering Building, Room 238

For more information please visit University of Washington Department of Mathematics