**May 31, 2010 **

Technische Universität Berlin

Fakultät II, Institut für Mathematik

Str. des 17. Juni 136,

10623 Berlin

room MA 041

** Lecture - 14:15**

*Abstract:*

We establish the existence of free energy limits for several sparse
random hypergraph models corresponding to certain combinatorial models
on Erd{\"o}s-R\'{e}nyi graph $G(N,c/N)$ and random $r$-regular graph
$G(N,r)$.
For a variety of models, including independent sets, MAX-CUT, Coloring
and K-SAT, we prove that the free energy both at a positive and zero
temperature, appropriately rescaled, converges to a limit as the size
of the underlying graph diverges to infinity. In the zero temperature
case, this is interpreted as the existence of the scaling limit for
the corresponding combinatorial optimization problem.
For example, as a special case we prove that the size of a largest
independent set in these graphs, normalized by the number of nodes
converges to a limit w.h.p., thus resolving an open problem mentioned
by several experts: Wormald '99, Aldous-Steele 2003,
Bollob\'as-Riordan 2008, as well as Janson-Thomason 2008.

Our approach is based on extending and simplifying the
Guerra-Toninelli interpolation method, as well as extending the work
of Franz-Leone and Montanari. We provide a simpler combinatorial
approach and work with the zero temperature case (optimization)
directly both in the case of Erd{\"o}s-R\'{e}nyi graph $G(N,c/N)$ and
random regular graph $G(N,r)$, while the previous authors handled the
zero temperature case (for other models) by taking limits of positive
temperature models.
In addition we establish the large deviations principle for the
satisfiability property for constraint satisfaction problems such as
Coloring, K-SAT and NAE-K-SAT.

This is joint work with D. Gamarnik (MIT) and M. Bayati (Stanford).

**Colloquium - 16:00**

*Abstract:*

Let S be a finite point set in R^d, and let epsilon < 1 be a parameter.
A "weak epsilon-net" for S ("for convex sets") is another set of points N which
intersects the convex hull of every subset of S of size at least epsilon*|S|.
It is known that the size of N need only depend on epsilon (and on d), but not
on |S|. Let r = 1/epsilon.

We examine a special case where the given set S is "intrinsically one-dimensional",
namely when S lies on a convex curve. (A convex curve in R^d is a curve that intersects
every hyperplane at most d times, e.g. the moment curve..) We improve the previous bounds
for this case (which were of the form O(r * polylog(r))), and prove that in this case there
exists a weak 1/r-net N of size O(r * 2^poly(alpha(r))), where alpha(x) is the extremely
slow-growing inverse Ackermann function.

We also prove a lower bound: For every fixed d>=3 and every r>1 there exists an S lying
on a convex curve, for which every weak 1/r-net has size Omega(r * 2^poly(alpha(r))).
The difference between the upper and the lower bound is that in the former, the polynomial
in the exponent is of degree ~d^2/4, while in the latter it is of degree ~d/2.

The set S that achieves the lower bound is actually very simple: It is just a point set suitably "stretched out".

Joint work with Noga Alon, Boris Bukh, Haim Kaplan, Jiri Matousek, Micha Sharir, and Shakhar Smorodinsky.

Letzte Aktualisierung:
06.05.2010