CompDP: A Framework for Simultaneous Subgraph Counting Under Connectivity Constraints

The subgraph counting problem computes the number of subgraphs of a given graph that satisfy some constraints. Among various constraints imposed on a graph, those regarding the connectivity of vertices, such as "these two vertices must be connected," have great importance since they are indispensable for determining various graph substructures, e.g., paths, Steiner trees, and rooted spanning forests. In this view, the subgraph counting problem under connectivity constraints is also important because counting such substructures often corresponds to measuring the importance of a vertex in network infrastructures. However, we must solve the subgraph counting problems multiple times to compute such an importance measure for every vertex. Conventionally, they are solved separately by constructing decision diagrams such as BDD and ZDD for each problem. However, even solving a single subgraph counting is a computationally hard task, preventing us from solving it multiple times in a reasonable time. In this paper, we propose a dynamic programming framework that simultaneously counts subgraphs for every vertex by focusing on similar connectivity constraints. Experimental results show that the proposed method solved multiple subgraph counting problems about 10-20 times faster than the existing approach for many problem settings

Efficient Sampling and Counting of Graph Structures related to Chordal Graphs

Counting problems aim to count the number of solutions for a given input, for example, counting the number of variable assignments that satisfy a Boolean formula. Sampling problems aim to produce a random object from a desired distribution, for example, producing a variable assignment drawn uniformly at random from all assignments that satisfy a Boolean formula. The problems of counting and sampling of graph structures on different types of graphs have been studied for decades for their great importance in areas like complexity theory and statistical physics. For many graph structures such as independent sets and acyclic orientations, it is widely believed that no exact or approximate (with an arbitrarily small error) polynomial-time algorithms on general graphs exist. Therefore, the research community studies various types of graphs, aiming either to design a polynomial-time counting or sampling algorithm for such inputs, or to prove a corresponding inapproximability result. Chordal graphs have been studied widely in both AI and theoretical computer science, but their study from the counting perspective has been relatively limited. Previous works showed that some graph structures can be counted in polynomial time on chordal graphs, when their counting on general graphs is provably computationally hard. The main objective of this thesis is to design and analyze counting and sampling algorithms for several well-known graph structures, including independent sets and different types of graph orientations, on chordal graphs. Our contributions can be described from two perspectives: evaluating the performances of some well-known sampling techniques, such as Markov chain Monte Carlo, on chordal graphs; and showing that the chordality does make those counting problems polynomial-time solvable

Zuverlässigkeit in gerichteten Netzwerken

Diese Arbeit beschäftigt sich mit verschiedenen Zuverlässigkeitsproblemen in gerichteten Netzwerken. Dabei wird speziell die s,t-Zuverlässigkeit und die s,T-Zuverlässigkeit betrachtet. Dazu werden verschiedene Berechnungs- und Reduktionsmöglichkeiten vorgestellt und anhand von Testrechnungen miteinander verglichen. Außerdem werden für spezielle Graphenklassen explizite und rekursive Formeln angegeben

Combinatorial and graph theoretical aspects of two-edge connected reliability

Die Untersuchung von Zuverlässigkeitsnetzwerken geht bis zum frühen 20. Jahrhundert zurück. Diese Arbeit beschäftigt sich hauptsächlich mit der Zweifach-Kantenzusammenhangswahrscheinlichkeit. Zuerst werden einfache Algorithmen, die aber für allgemeine Graphen nicht effizient sind, gezeigt, zusammen mit Reduktionen. Weiterhin werden Charakterisierungen von Kanten bezogen auf Wegemengen gezeigt. Neue strukturelle Bedingungen für diese werden vorgestellt. Neue Ergebnisse liegen ebenfalls für Graphen hoher Dichte und Symmetrie vor, genauer für vollständige und vollständig bipartite Graphen. Naturgemäß sind Graphen von geringer Dichte hier einfacher in der Untersuchung. Die Arbeit zeigt Ergebnisse für Kreise, Räder und Leiterstrukturen. Graphen mit beschränkter Weg- beziehungsweise Baumweite haben polynomiale Algorithmen und in Spezialfällen einfache Formeln, die ebenfalls vorgestellt werden. Der abschließende Teil beschäftigt sich mit Schranken und Approximationen

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

Paths and walks, forests and planes : arcadian algorithms and complexity

This dissertation is concerned with new results in the area of parameterized algorithms and complexity. We develop a new technique for hard graph problems that generalizes and unifies established methods such as Color-Coding, representative families, labelled walks and algebraic fingerprinting. At the heart of the approach lies an algebraic formulation of the problems, which is effected by means of a suitable exterior algebra. This allows us to estimate the number of simple paths of given length in directed graphs faster than before. Additionally, we give fast deterministic algorithms for finding paths of given length if the input graph contains only few of such paths. Moreover, we develop faster deterministic algorithms to find spanning trees with few leaves. We also consider the algebraic foundations of our new method. Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways. Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis. Secondly, we prove that counting forests with a given number of edges is at least as hard as counting cliques of a given size.Diese Dissertation befasst sich mit neuen Ergebnissen auf dem Gebiet parametrisierter Algorithmen und Komplexitätstheorie. Wir entwickeln eine neue Technik für schwere Graphprobleme, die etablierte Methoden wie Color-Coding, representative families, labelled walks oder algebraic fingerprinting verallgemeinert und vereinheitlicht. Kern der Herangehensweise ist eine algebraische Formulierung der Probleme, die vermittels passender Graßmannalgebren geschieht. Das erlaubt uns, die Anzahl einfacher Pfade gegebener Länge in gerichteten Graphen schneller als bisher zu schätzen. Außerdem geben wir schnelle deterministische Verfahren an, Pfade gegebener Länge zu finden, falls der Eingabegraph nur wenige solche Pfade enthält. Übrigens entwickeln wir schnellere deterministische Algorithmen, um Spannbäume mit wenigen Blättern zu finden. Wir studieren außerdem die algebraischen Grundlagen unserer neuen Methode. Weiters untersuchen wir die fine-grained-Komplexität davon, die genaue Anzahl von Wäldern einer gegebenen Kantenzahl in einem gegebenen ungerichteten Graphen zu bestimmen. Und zwar erfolgt das auf zwei verschiedene Arten. Erstens vervollständigen wir die Komplexitätsklassifizierung der Tutte-Ebene unter Annahme der Expo- nentialzeithypothese. Zweitens beweisen wir, dass Wälder mit gegebener Kantenzahl zu zählen, wenigstens so schwer ist, wie Cliquen gegebener Größe zu zählen.Cluster of Excellence (Multimodal Computing and Interaction