## Tuesday 10 November 2009

An Oxford-Warwick Joint Seminar Afternoon will be held in the Mathematical Institute, University of Oxford on Tuesday 10 November 2009.  The schedule is as follows:

2.00 pm                    Harald Raecke (Warwick)

Oblivious Routing in the L_p norm

2.50 pm                    Colin McDiarmid (Oxford)
Random graphs with few disjoint cycles

3.40 pm                     Tea

4.30 pm                     Gregory Sorkin (IBM Research, NY)

The Power of Choice in a Generalized Polya Urn Model

The first two talks will be in L3 and the third in SR2.  Abstracts can be found below.  Further details can be obtained from Alex Scott (contact form).

Harald Raecke (Warwick), Oblivious Routing in the L_p norm

Gupta et al. introduced a very general multi-commodity flow problem in which the cost of a given flow solution on a graph G=(V,E) is calculated by first computing the link loads via a load-function l, that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual link loads into a single number via an aggregation function.

We show the existence of an oblivious routing scheme with competitive ratio O(log n) and a lower bound of Omega(log n/loglog n) for this model when the aggregation function agg is an L_p-norm.

Our results can also be viewed as a generalization of the work on approximating metrics by a distribution over dominating tree metrics and the work on minimum congestion oblivious. We provide a convex combination of trees such that routing according to the tree distribution approximately minimizes the L_p-norm of the link loads. The embedding techniques of Bartal and Fakcharoenphol et al. [FRT03] can be viewed as solving this problem in the L_1-norm while the result on congestion minmizing oblivious routing solves it for L_\infty. We give a single proof that shows the existence of a good tree-based oblivious routing for any L_p-norm.

Colin McDiarmid (Oxford), Random graphs with few disjoint cycles

Fix a positive integer $k$, and consider the class of all graphs which do not have $k+1$  vertex-disjoint cycles.  A classical result of Erd\H{o}s and P\'{o}sa says that each such graph $G$ contains a blocker of size at most $f(k)$.  Here a {\em blocker} is a set $B$ of vertices such that $G-B$ has no cycles.

We give a minor extension of this result, and deduce that almost all such labelled graphs on vertex set $1,\ldots,n$ have a blocker of size $k$.  This yields an asymptotic counting formula for such graphs; and allows us to deduce further properties of a graph $R_n$ taken uniformly at random from the class: we see for example that the probability that $R_n$ is connected tends to a specified limit as $n \to \infty$.

There are corresponding results when we consider unlabelled graphs with few disjoint cycles. We consider also variants of the problem involving for example disjoint long cycles.

This is joint work with Valentas Kurauskas and Mihyun Kang.

Gregory Sorkin (IBM Research, NY), The Power of Choice in a Generalized Polya Urn Model

We introduce a "Polya choice" urn model combining elements of the well known "power of two choices" model and the "rich get richer" model. From a set of $k$ urns, randomly choose $c$ distinct urns with probability proportional to the product of a power $\gamma>0$ of their occupancies, and increment one with the smallest occupancy. The model has an interesting phase transition. If $\gamma \leq 1$, the urn occupancies are asymptotically equal with probability 1. For $\gamma>1$, this still occurs with positive probability, but there is also positive probability that some urns get only finitely many balls while others get infinitely many.