## Subsampling Mathematical Relaxations and Average-case Complexity

We consider the following two questions:

Boaz Barak

Institute for Advanced Study

May 24, 2010

We consider the following two questions:

Madhur Tulsiani

Institute for Advanced Study

May 18, 2010

The small-set expansion conjecture introduced by Raghavendra and Steuerer is a natural hardness assumption concerning the problem of approximating edge expansion of small sets (of size $\delta n$) in graphs. It was shown to be intimately connected to the well-known Unique Games Conjecture.

Pursuing this line of research further, we obtain the following results:

Amir Yehudayoff

Institute for Advanced Study

May 11, 2010

Umberto Zannier

Scuola Normale Superiore de Pisa, Italy

May 4, 2010

Dana Moshkovitz

Institute for Advanced Study

April 27, 2010

Vijay Vazirani

Georgia Institute of Technology

April 19, 2010

*It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest. Each participant in a competitive economy is led by an invisible hand to promote an end which was no part of his intention.*

Adam Smith, 1776

Eyal Lubetzky

Microsoft Research Redmond

April 13, 2010

James Lee

University of Washington

April 12, 2010

The cover time of a graph is one of the most basic and well-studied properties of the simple random walk, and yet a number of fundamental questions concerning cover times have remained open. We show that there is a deep connection between cover times of graphs and Talagrand's majorizing measure theory. In particular, we prove that the cover time can be characterized, up to universal constants, by the majorizing measure value of a certain metric space on the underlying graph.

Robert MacPherson

Institute for Advanced Study

April 7, 2010

The ordinary homology of a subset S of Euclidean space depends only on its topology. By systematically organizing homology of neighborhoods of S, we get quantities that measure the shape of S, rather than just its topology. These quantities can be used to define a new notion of fractional dimension of S. They can also be effectively calculated on a computer.

Mark Braverman

Microsoft Research New England

April 6, 2010