Forthcoming Seminars › Seminar Series

Combinatorial Theory Seminar

Tue, 09/10/2007
14:30 		Bruce Reed (McGill/INRIA/CNRS)) Combinatorial Theory Seminar Add to calendar L3
Tying down the diameter of G(n,p).
We show that the diameter of G(n,p) is concentrated on one of three values provided the average degree p(n-1) goes to infity with n. This is joint work with N. Fountoulakis even though he refuses to admit it.
Tue, 16/10/2007
14:30 		Charles Semple (University of Canterbury, NZ) Combinatorial Theory Seminar Add to calendar L3
Combinatorial Problems in Conservation Biology
A central task in conservation biology is measuring, predicting, and preserving biological diversity as species face extinction. Dating back to 1992, phylogenetic diversity is a prominent notion for measuring the biodiversity of a collection of species. This talk gives a flavour of some the combinatorial and algorithmic problems and recent solutions associated with computing this measure. This is joint work with Magnus Bordewich (Durham University, UK) and Andreas Spillner (University of East Anglia, UK).
Tue, 16/10/2007
16:30 		Nicolas Broutin (McGill) Combinatorial Theory Seminar Add to calendar SR1
The structure and profile of digital trees
Digital trees is a general structure to manipulate sequences of characters. We propose a novel approach to the structure of digital trees. It shades some new light on the profile of digital trees, and provides a unified explanation of the relationships between different kinds of digital trees. The idea relies on the distinction of nodes based on their type, i.e., the set of their children. Only two types happen to matter when studying the number of nodes lying at a specified level: the nodes with a full set of children which constitutes the core, and the nodes with a single child producing spaghetti-like trees hanging down the core. We will explain the distinction and its applications on a number of examples related to data structures such as the TST of Bentley and Sedgewick. This is joint work with Luc Devroye.
Tue, 23/10/2007
16:30 		Laszlo Szekely (USC) Combinatorial Theory Seminar Add to calendar SR1
A new tool for asymptotic enumeration: the Lovasz Local Lemma?
The Lovasz Local Lemma is known to have an extension for cases where the dependency graph requirement is relaxed to negative dependency graph (Erdos-Spencer 1991). The difficulty is to find relevant negative dependency graphs that are not dependency graphs. We provide two generic constructions for negative dependency graphs, in the space of random matchings of complete and complete bipartite graphs. As application, we prove existence results for hypergraph packing and Turan type extremal problems. We strengthen the classic probabilistic proof of Erdos for the existence of graphs with large girth and chromatic number by prescribing the degree sequence, which has to satisfy some mild conditions. A more surprising application is that tight asymptotic lower bounds can be obtained for asymptotic enumeration problems using the Lovasz Local Lemma. This is joint work with Lincoln Lu.
Tue, 30/10/2007
13:30 		Matthias Reitzner (Vienna) Combinatorial Theory Seminar Add to calendar L3
Random polytopes
Let $K \subset {\mathbb R}^d$ be a convex set. Choose $n$ random points in $K$, and denote by $P_n$ their convex hull. We call $P_n$ a random polytope. Investigations concerning the expected value of functionals of $P_n$, like volume, surface area, and number of vertices, started in 1864 with a problem raised by Sylvester and now are a classical part of stochastic and convex geometry. The last years have seen several new developments about distributional aspects of functionals of random polytopes. In this talk we concentrate on these recent results such as central limit theorems and tail inequalities, as the number of random points tends to infinity.
Tue, 30/10/2007
15:30 		Pierre Charbit (Paris) Combinatorial Theory Seminar Add to calendar L3
Infinite locally random graphs
The Rado Graph R is a graph on a countably infinite number of vertices that can be characterized by the following property: for every pair of finite, disjoint subsets of vertices X and Y, there exists a vertex that is adjacent to every vertex in X and none in Y. It is often called the Random Graph for the following reason: for any 0
Tue, 06/11/2007
13:30 		Oliver Riordan (Oxford) Combinatorial Theory Seminar Add to calendar L3
The diameter of G9n,p) via branching processes
One of the main tools in studying sparse random graphs with independence between different edges is local comparison with branching processes. Recently, this method has been used to determine the asymptotic behaviour of the diameter (largest graph distance between two points that are in the same component) of various sparse random graph models, giving results for $ G(n,c/n) $ as special cases. Nick Wormald and I have applied this method to $ G(n,c/n) $ itself, obtaining a much stronger result, with a best-possible error term. We also obtain results as $ c $ varies with $ n $, including results almost all the way down to the phase transition.
Tue, 06/11/2007
15:30 		Tobias Muller (Eindhoven) Combinatorial Theory Seminar Add to calendar SR1
First order properties of random graphs
A graph property is a first order property if it can be written as a logic sentence with variables ranging over the vertices of the graph. A sequence of random graphs (G_n)_n satisfies the zero-one law if the probability that G_n satisfies P tends to either zero or one for every first order property P. This is for instance the case for G(n,p) if p is fixed. I will survey some of the most important results on the G(n,p)-model and then proceed to discuss some work in progress on other graph models.
Tue, 13/11/2007
13:30 		Leen Stougie (Einhoven) Combinatorial Theory Seminar Add to calendar L3
A Linear Bound on the Diameter of the transportation Polytope
The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [Hitchcock, 1941] and independently by Koopmans in 1947 [Koopmans, 1948], and appears in any standard introductory course on operations research. The mxn TP has m supply points and n demand points. Each supply Point i holds a quantity r_i, and each demand point j wants a quantity c_j, with the sum of femands equal to the sum of supplies. A solution to the problem can be written as a mxn matrix X with entries decision x_{ij} having value equal to the amount transported from supply point i to demand point j. The objective is to minimize total transportation costs when unit transporation costs between each supply and each demand point are given. The set of feasible solutions of TP, is called the transportation polytope. The 1-skeleton (edge graph) of this polytope is defined as the graph with vertices the vertices of the polytope and edges its 1-dimensional faces. In 1957 W.M. Hirsch stated his famous conjecture cf. [Dantzig, 1963]) saying that any d-dimensional polytope with n facets has diameter at most n-d. So far the best bound for any polytope is O(n^{\log d+1}) [Kalai and Kleitman, 1992]. Any strongly polynomial bound is still lacking. Such bounds have been proved for some special classes of polytopes (for examples, see [Schrijver, 1995]). Among those are some special classes of transportation polytopes [Balinski, 1974],[Bolker, 1972] and the polytope of the dual of TP [Balinski, 1974]. The first strongly polynomial bound on the diameter of the transportation polytope was given by Dyer and Frieze [DyerFrieze, 1994]. Actually, they prove a bound on the diameter of any polytope {x|Ax=b} where A is a totally unimodular matrix. The proof is complicated and indirect, using the probabilistic method. Moreover, the bound is huge O(m^{16}n^3ln(mn))3) assuming m less than or equal to n. We will give a simple proof that the diameter of the transportation polytope is less than 8(m+n-2). The proof is constructive: it gives an algorithm that describes how to go from any vertex to any other vertex on the transportation polytope in less than 8(m+n-2) steps along the edges. According to the Hirsch Conjecture the bound on the TP polytope should be m+n-1. Thus we are within a multiplicative factor 8 of the Hirsch bound. Recently C. Hurkens refined our analysis and diminished the bound by a factor 2, arriving at 4(m+n-2). I will indicate the way he achieved this as well.
Tue, 13/11/2007
15:30 		Rob Morris (Cambridge) Combinatorial Theory Seminar Add to calendar SR1
Bootstrap percolation and the Ising model
Glauber dynamics on $ \mathbb{Z}^d $ is a dynamic representation of the zero-temperature Ising model, in which the spin (either $ + $ or $ - $) of each vertex updates, at random times, to the state of the majority of its neighbours. It has long been conjectured that the critical probability $ p_c(\mathbb{Z}^d) $ for fixation (every vertex eventually in the same state) is $ 1/2 $, but it was only recently proved (by Fontes, Schonmann and Sidoravicius) that $ p_c(\mathbb{Z}^d) < 1 $. Bootstrap percolation is a simpler, monotone version of Glauber dynamics, in which sites can only change state in one direction (from $ - $ to $ + $, say). In this talk I shall discuss some recent advances in the study of bootstrap percolation on $ [n]^d $, and show how these can be applied to obtain improved bounds on the critical probability for Glauber dynamics in high dimensions. This is joint work with József Balogh and Béla Bollobás.
Tue, 20/11/2007
13:30 		Georg Gottlob (Oxford) Combinatorial Theory Seminar Add to calendar L3
Minimal hypergraph transversals and their use in Computer Science
Hypergraph Transversals have been studied in Mathematics for a long time (e.g. by Berge) . Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science, especially in database Theory, Logic, and AI. We give a survey of various applications and review some recent results on the complexity of computing all minimal transversals of a given hypergraph.
Tue, 20/11/2007
15:30 		Sebastian Muller (Graz) Combinatorial Theory Seminar Add to calendar SR1
Transcience and recurrence for branching random walks in random environment
We give different criteria for transience of branching Markov chains. These conditions enable us to give a classification of branching random walks in random environment (BRWRE) on Cayley graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $ \Z^d. $ Furthermore, we emphasize the interplay between branching Markov chains, the spectral radius, and some generating functions.
Tue, 27/11/2007
13:30 		Mike Steel (University of Canterbury, NZ) Combinatorial Theory Seminar Add to calendar L3
Combinatorial approaches in phylogenetics
Phylogenetics is the reconstruction and analysis of 'evolutionary' trees and graphs in biology (and related areas of classification, such as linguistics). Discrete mathematics plays an important role in the underlying theory. We will describe some of the ways in which concepts from combinatorics (e.g. poset theory, greedoids, cyclic permutations, Menger's theorem, closure operators, chordal graphs) play a central role. As well as providing an overview, we also describe some recent and new results, and outline some open problems.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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