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Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Random combinatorial structures and randomized search heuristics
This thesis is concerned with the probabilistic analysis of random combinatorial structures and the runtime analysis of randomized search heuristics.
On the subject of random structures, we investigate two classes of combinatorial objects. The first is the class of planar maps and the second is the class of generalized parking functions. We identify typical properties of these structures and show strong concentration results on the probabilities that these properties hold. To this end, we develop and apply techniques based on exact enumeration by generating functions. For several types of random planar maps, this culminates in concentration results for the degree sequence. For parking functions, we determine the distribution of the defect, the most characteristic parameter. On the subject of randomized search heuristics, we present, improve, and unify different probabilistic methods and their applications. In this, special focus is given to potential functions and the analysis of the drift of stochastic processes. We apply these techniques to investigate the runtimes of evolutionary algorithms. In particular, we show for several classical problems in combinatorial optimization how drift analysis can be used in a uniform way to give bounds on the expected runtimes of evolutionary algorithms.Diese Dissertationsschrift beschĂ€ftigt sich mit der wahrscheinlichkeitstheoretischen Analyse von zufĂ€lligen kombinatorischen Strukturen und der Laufzeitanalyse randomisierter Suchheuristiken. Im Bereich der zufĂ€lligen Strukturen untersuchen wir zwei Klassen kombinatorischer Objekte. Dies sind zum einen die Klasse aller kombinatorischen Einbettungen planarer Graphen und zum anderen eine Klasse diskreter Funktionen mit bestimmten kombinatorischen Restriktionen (generalized parking functions). FĂŒr das Studium dieser Klassen entwickeln und verwenden wir zĂ€hlkombinatorische Methoden die auf erzeugenden Funktionen basieren. Dies erlaubt uns, Konzentrationsresultate fĂŒr die Gradsequenzen verschiedener Typen zufĂ€lliger kombinatorischer Einbettungen planarer Graphen zu erzielen. DarĂŒber hinaus erhalten wir Konzentrationsresultate fĂŒr den charakteristischen Parameter, den Defekt, zufĂ€lliger Instanzen der untersuchten diskreten Funktionen.
Im Bereich der randomisierten Suchheuristiken prÀsentieren und erweitern wir verschiedene wahrscheinlichkeitstheoretische Methoden der Analyse. Ein besonderer Fokus liegt dabei auf der Analyse der Drift stochastischer Prozesse. Wir wenden diese Methoden in der Laufzeitanalyse evolutionÀrer Algorithmen an. Insbesondere zeigen wir, wie mit Hilfe von Driftanalyse die erwarteten Laufzeiten evolutionÀrer Algorithmen auf verschiedenen klassischen Problemen der kombinatorischen Optimierung auf einheitliche Weise abgeschÀtzt werden können
Vertices of Degree k in Random Maps â
This work is devoted to the study of the typical structure of a random map. Maps are planar graphs embedded in the plane. We investigate the degree sequences of random maps from families of a certain type, which, among others, includes fundamental map classes like those of biconnected maps, 3-connected maps, and triangulations. In particular, we develop a general framework that allows us to derive relations and exact asymptotic expressions for the expected number of vertices of degree k in random maps from these classes, and also provide accompanying large deviation statements. Extending the work of Gao and Wormald (Combinatorica, 2003) on random general maps, we obtain as results of our framework precise information about the number of vertices of degree k in random biconnected, 3-connected, loopless, and bridgeless maps.