Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Erdős-Hajnal Conjecture for Graphs with Bounded VC-Dimension

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn)1-o(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, eclogn, due to Erds and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erds-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e(logn). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the ultra-strong regularity lemma for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)O(d), improving the original bound of (1/epsilon)O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Turan numbers for graphs with bounded VC-dimension

Computational methods in protein structure comparison and analysis of protein interaction networks

Proteins are versatile biological macromolecules that perform numerous functions in a living organism. For example, proteins catalyze chemical reactions, store and transport various small molecules, and are involved in transmitting nerve signals. As the number of completely sequenced genomes grows, we are faced with the important but daunting task of assigning function to proteins encoded by newly sequenced genomes. In this thesis we contribute to this effort by developing computational methods for which one use is to facilitate protein function assignment. Functional annotation of a newly discovered protein can often be transferred from that of evolutionarily related proteins of known function. However, distantly related proteins can still only be detected by the most accurate protein structure alignment methods. As these methods are computationally expensive, they are combined with less accurate but fast methods to allow large-scale comparative studies. In this thesis we propose a general framework to define a family of protein structure comparison methods that reduce protein structure comparison to distance computation between high-dimensional vectors and therefore are extremely fast. Interactions among proteins can be detected through the use of several mature experimental techniques. These interactions are routinely represented by a graph, called a protein interaction network, with nodes representing the proteins and edges representing the interactions between the proteins. In this thesis we present two computational studies that explore the connection between the topology of protein interaction networks and protein biological function. Unfortunately, protein interaction networks do not explicitly capture an important aspect of protein interactions, their dynamic nature. In this thesis, we present an automatic method that relies on graph theoretic tools for chordal and cograph graph families to extract dynamic properties of protein interactions from the network topology. An intriguing question in the analysis of biological networks is whether biological characteristics of a protein, such as essentiality, can be explained by its placement in the network. In this thesis we analyze protein interaction networks for Saccharomyces cerevisiae to identify the main topological determinant of essentiality and to provide a biological explanation for the connection between the network topology and essentiality

Problems in graph theory and partially ordered sets

This dissertation answers problems in three areas of combinatorics - processes on graphs, graph coloring, and antichains in a partially ordered set.First we consider Zero Forcing on graphs, an iterative infection process introduced by AIM Minimum Rank - Special Graphs Workgroup in 2008. The Zero Forcing process is a graph infection process obeying the following rules: a white vertex is turned black if it is the only white neighbor of some black vertex. The Zero Forcing Number of a graph is the minimum cardinality over all sets of black vertices such that, after a finite number of iterations, every vertex is black. We establish some results about the zero forcing number of certain graphs and provide a counter example of a conjecture of Gentner and Rautenbach. This chapter is joint with Gabor Meszaros, Antonio Girao, and Chapter 3 appears in Discrete Math, Vol. 341(4).In the second part, we consider problems in the area of Dynamic Coloring of graphs. Originally introduced by Montgomery in 2001, the r-dynamic chromatic number of a graph G is the least k such that V(G) is properly colored, and each vertex is adjacent to at least r different colors. In this coloring regime, we prove some bounds for graphs with lattice like structures, hypercubes, generalized intervals, and other graphs of interest. Next, we establish some of the first results in the area of r-dynamic coloring on random graphs. The work in this section is joint with Peter van Hintum.In the third part, we consider a question about the structure of the partially ordered set of all connected graphs. Let G be the set of all connected graphs on vertex set [n]. Define the partial ordering \u3c on G as follows: for G,H G let G \u3c H if E(G) E(H). The poset (G

High-order renormalization of scalar quantum fields

Thema dieser Dissertation ist die Renormierung von perturbativer skalarer Quantenfeldtheorie bei großer Schleifenzahl. Der Hauptteil der Arbeit ist dem Einfluss von Renormierungsbedingungen auf renormierte Greenfunktionen gewidmet. Zunächst studieren wir Dyson-Schwinger-Gleichungen und die Renormierungsgruppe, inklusive der Gegenterme in dimensionaler Regularisierung. Anhand zahlreicher Beispiele illustrieren wir die verschiedenen Größen. Alsdann diskutieren wir, welche Freiheitsgrade ein Renormierungsschema hat und wie diese mit den Gegentermen und den renormierten Greenfunktionen zusammenhängen. Für ungekoppelte Dyson-Schwinger-Gleichungen stellen wir fest, dass alle Renormierungsschemata bis auf eine Verschiebung des Renormierungspunktes äquivalent sind. Die Verschiebung zwischen kinematischer Renormierung und Minimaler Subtraktion ist eine Funktion der Kopplung und des Regularisierungsparameters. Wir leiten eine neuartige Formel für den Fall einer linearen Dyson-Schwinger Gleichung vom Propagatortyp her, um die Verschiebung direkt aus der Mellintransformation des Integrationskerns zu berechnen. Schließlich berechnen wir obige Verschiebung störungstheoretisch für drei beispielhafte nichtlineare Dyson-Schwinger-Gleichungen und untersuchen das asymptotische Verhalten der Reihenkoeffizienten. Ein zweites Thema der vorliegenden Arbeit sind Diffeomorphismen der Feldvariable in einer Quantenfeldtheorie. Wir präsentieren eine Störungstheorie des Diffeomorphismusfeldes im Impulsraum und verifizieren, dass der Diffeomorphismus keinen Einfluss auf messbare Größen hat. Weiterhin untersuchen wir die Divergenzen des Diffeomorphismusfeldes und stellen fest, dass die Divergenzen Wardidentitäten erfüllen, die die Abwesenheit dieser Terme von der S-Matrix ausdrücken. Trotz der Wardidentitäten bleiben unendlich viele Divergenzen unbestimmt. Den Abschluss bildet ein Kommentar über die numerische Quadratur von Periodenintegralen.This thesis concerns the renormalization of perturbative quantum field theory. More precisely, we examine scalar quantum fields at high loop order. The bulk of the thesis is devoted to the influence of renormalization conditions on the renormalized Green functions. Firstly, we perform a detailed review of Dyson-Schwinger equations and the renormalization group, including the counterterms in dimensional regularization. Using numerous examples, we illustrate how the various quantities are computable in a concrete case and which relations they satisfy. Secondly, we discuss which degrees of freedom are present in a renormalization scheme, and how they are related to counterterms and renormalized Green functions. We establish that, in the case of an un-coupled Dyson-Schwinger equation, all renormalization schemes are equivalent up to a shift in the renormalization point. The shift between kinematic renormalization and Minimal Subtraction is a function of the coupling and the regularization parameter. We derive a novel formula for the case of a linear propagator-type Dyson-Schwinger equation to compute the shift directly from the Mellin transform of the kernel. Thirdly, we compute the shift perturbatively for three examples of non-linear Dyson-Schwinger equations and examine the asymptotic growth of series coefficients. A second, smaller topic of the present thesis are diffeomorphisms of the field variable in a quantum field theory. We present the perturbation theory of the diffeomorphism field in momentum space and find that the diffeomorphism has no influence on measurable quantities. Moreover, we study the divergences in the diffeomorphism field and establish that they satisfy Ward identities, which ensure their absence from the S-matrix. Nevertheless, the Ward identities leave infinitely many divergences unspecified and the diffeomorphism theory is perturbatively unrenormalizable. Finally, we remark on a third topic, the numerical quadrature of Feynman periods

Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations

In the first purpose, we concentrate on the theory of quantum integrable systems underlying the Connes-Kreimer approach. We introduce a new family of Hamiltonian systems depended on the perturbative renormalization process in renormalizable theories. It is observed that the renormalization group can determine an infinite dimensional integrable system such that this fact provides a link between this proposed class of motion integrals and renormalization flow. Moreover, with help of the integral renormalization theorems, we study motion integrals underlying Bogoliubv character and BCH series to obtain a new family of fixed point equations. In the second goal, we consider the combinatorics of Connes-Marcolli approach to provide a Hall rooted tree type reformulation from one particular object in this theory namely, universal Hopf algebra of renormalization $H_{\mathbb{U}}$. As the consequences, interesting relations between this Hopf algebra and some well-known combinatorial Hopf algebras are obtained and also, one can make a new Hall polynomial representation from universal singular frame such that based on the universal nature of this special loop, one can expect a Hall tree type scattering formula for physical information such as counterterms. In the third aim, with attention to the given rooted tree version of $H_{\mathbb{U}}$ and by applying the Connes-Marcolli's universal investigation, we discover a new geometric explanation from Dyson-Schwinger equations.Comment: Keywords: Combinatorial Hopf Algebras; Connes-Kreimer Renormalization Group; Connes-Kreimer-Marcolli Perturbative Renormalization; Dyson-Schwinger Equations; Hall Rooted Trees; Quantum Integrable Systems; Renormalizable Quantum Field Theory; Riemann-Hilbert Correspondence; Universal Hopf Algebra of Renormalizatio