Subdivisions with Distance Constraints in Large Graphs

In this dissertation we are concerned with sharp degree conditions that guarantee the existence of certain types of subdivisions in large graphs. Of particular interest are subdivisions with a certain number of arbitrarily specified vertices and with prescribed path lengths. Our non-standard approach makes heavy use of the Regularity Lemma (Szemerédi, 1978), the Blow-Up Lemma (Komlós, Sárkózy, and Szemerédi, 1994), and the minimum degree panconnectivity criterion (Williamson, 1977).Sharp minimum degree criteria for a graph G to be H-linked have recently been discovered. We define (H,w,d)-linkage, a condition stronger than H-linkage, by including a weighting function w consisting of required lengths for each edge-path of a desired H-subdivision. We establish sharp minimum degree criteria for a large graph G to be (H,w,d)-linked for all nonnegative d. We similarly define the weaker condition (H,S,w,d)-semi-linkage, where S denotes the set of vertices of H whose corresponding vertices in an H-subdivision are arbitrarily specified. We prove similar sharp minimum degree criteria for a large graph G to be (H,S,w,d)-semi-linked for all nonnegativeWe also examine path coverings in large graphs, which could be seen as a special case of (H,S,w)-semi-linkage. In 2000, Enomoto and Ota conjectured that a graph G of order n with degree sum σ2(G) satisfying σ2(G) \u3e n + k - 2 may be partitioned into k paths, each of prescribed order and with a specified starting vertex. We prove the Enomoto-Ota Conjecture for graphs of sufficiently large order

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.Comment: To appear in SIAM J. Computin

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