Uri Feige, Weizmann Institute
Preprocessing of NPhard problems
Abstract:
Consider a setting in which an input instance for an NPhard optimization problem is supplied in two steps. In the first step, one gets to see some
partial information about the input instance, referred to as a "preview". Based on this preview, a preprocessing algorithm can spend arbitrary (e.g., exponential)
time in preparing some polynomial size "advice" string. In the second step, one gets to see the full input instance. Thereafter, a polynomial time algorithm attempts
to solve the instance, and may use for this purpose the advice string prepared by the preprocessing algorithm. For various NPhard optimization problems we present
natural preview functions whose study appears to be well motivated. In certain cases we can prove that preprocessing leads to improved approximation ratios, and in
certain cases we can prove limitations on how much preprocessing can help. Many natural questions within this framework remain open, and can serve as fertile ground for future research.

Colin McDiarmid, University of Oxford
Random graphs from a minorclosed class
Abstract:
Consider a minorclosed class of graphs, such as the class of planar graphs. What can we say in general about a typical nvertex graph in the class, for large n?
For example, how likely is it to be connected? To answer such questions we need to be able to estimate the number a_n of such graphs. As a first step we want to know
that there is a growth constant, that is (a_n/n!)^{1/n} tends to a limit. We can say more if the class is smooth, that is a_n/na_{n1} tends to a limit.
It is known that if a minorclosed class is addable (has 2connected excluded minors) then it is smooth: we shall see for example that the leaffree graphs
in such a class also form a smooth class.

