Cyclic sums, network sharing and restricted edge cuts in graphs with long cycles

Cyclic Sums, Network Sharing and Restricted Edge Cuts in Graphs with Long Cycles Dieter Rautenbach , Lutz Volkmann Preprint series: 07-06, 8 MSC 2000 05A17 Partitions of integers 05C40 Connectivity Abstract We study graphs G = (V,E) containing a long cycle which for given integers a1, a2, ..., ak 2 N have an edge cut whose removal results in k components with vertex sets V1, V2, ..., Vk such that |Vi| ai for 1 i k. Our results closely relate to problems and recent research in network sharing and network reliability. Keywords: restricted edge connectivity, arbitrarily vertex decomposable graph, network reliability, network sharin

New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity

Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails

Desenvolvimentos da Conjetura de Fulkerson

Orientador: Christiane Neme CamposDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Em 1971, Fulkerson propôs a seguinte conjetura: todo grafo cúbico sem arestas de corte admite seis emparelhamentos perfeitos tais que cada aresta do grafo pertence a exatamente dois destes emparelhamentos. A Conjetura de Fulkerson tem desafiado pesquisadores desde sua publicação. Esta conjetura é facilmente verificada para grafos cúbicos 3-aresta-coloráveis. Portanto, a dificuldade do problema reside em estabelecer a conjetura para grafos cúbicos sem arestas de corte que não possuem 3-coloração de arestas. Estes grafos são chamados snarks. Nesta dissertação, a Conjetura de Fulkerson e os snarks são introduzidos com ¿ênfase em sua história e resultados mais relevantes. Alguns resultados relacionados à Conjetura de Fulkerson são apresentados, enfatizando suas conexões com outras conjeturas. Um breve histórico do Problema das Quatro Cores e suas relações com snarks também são apresentados. Na segunda parte deste trabalho, a Conjetura de Fulkerson é verificada para algumas famílias infinitas de snarks construídas com o método de Loupekine, utilizando subgrafos do Grafo de Petersen. Primeiramente, mostramos que a família dos LP0-snarks satisfaz a Conjetura de Fulkerson. Em seguida, generalizamos este resultado para a família mais abrangente dos LP1-snarks. Além disto, estendemos estes resultados para Snarks de Loupekine construídos com subgrafos de snarks diferentes do Grafo de PetersenAbstract: In 1971, Fulkerson proposed a conjecture that states that every bridgeless cubic graph has six perfect matchings such that each edge of the graph belongs to precisely two of these matchings. Fulkerson's Conjecture has been challenging researchers since its publication. It is easily verified for 3-edge-colourable cubic graphs. Therefore, the difficult task is to settle the conjecture for non-3-edge-colourable bridgeless cubic graphs, called snarks. In this dissertation, Fulkerson's Conjecture and snarks are presented with emphasis in their history and remarkable results. We selected some results related to Fulkerson's Conjecture, emphasizing their reach and connections with other conjectures. It is also presented a brief history of the Four-Colour Problem and its connections with snarks. In the second part of this work, we verify Fulkerson's Conjecture for some infinite families of snarks constructed with Loupekine's method using subgraphs of the Petersen Graph. More specifically, we first show that the family of LP0-snarks satisfies Fulkerson's Conjecture. Then, we generalise this result by proving that Fulkerson's Conjecture holds for the broader family of LP1-snarks. We also extend these results to even more general Loupekine Snarks constructed with subgraphs of snarks other than the Petersen GraphMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

Combinatorial Solutions for Shape Optimization in Computer Vision

This thesis aims at solving so-called shape optimization problems, i.e. problems where the shape of some real-world entity is sought, by applying combinatorial algorithms. I present several advances in this field, all of them based on energy minimization. The addressed problems will become more intricate in the course of the thesis, starting from problems that are solved globally, then turning to problems where so far no global solutions are known. The first two chapters treat segmentation problems where the considered grouping criterion is directly derived from the image data. That is, the respective data terms do not involve any parameters to estimate. These problems will be solved globally. The first of these chapters treats the problem of unsupervised image segmentation where apart from the image there is no other user input. Here I will focus on a contour-based method and show how to integrate curvature regularity into a ratio-based optimization framework. The arising optimization problem is reduced to optimizing over the cycles in a product graph. This problem can be solved globally in polynomial, effectively linear time. As a consequence, the method does not depend on initialization and translational invariance is achieved. This is joint work with Daniel Cremers and Simon Masnou. I will then proceed to the integration of shape knowledge into the framework, while keeping translational invariance. This problem is again reduced to cycle-finding in a product graph. Being based on the alignment of shape points, the method actually uses a more sophisticated shape measure than most local approaches and still provides global optima. It readily extends to tracking problems and allows to solve some of them in real-time. I will present an extension to highly deformable shape models which can be included in the global optimization framework. This method simultaneously allows to decompose a shape into a set of deformable parts, based only on the input images. This is joint work with Daniel Cremers. In the second part segmentation is combined with so-called correspondence problems, i.e. the underlying grouping criterion is now based on correspondences that have to be inferred simultaneously. That is, in addition to inferring the shapes of objects, one now also tries to put into correspondence the points in several images. The arising problems become more intricate and are no longer optimized globally. This part is divided into two chapters. The first chapter treats the topic of real-time motion segmentation where objects are identified based on the observations that the respective points in the video will move coherently. Rather than pre-estimating motion, a single energy functional is minimized via alternating optimization. The main novelty lies in the real-time capability, which is achieved by exploiting a fast combinatorial segmentation algorithm. The results are furthermore improved by employing a probabilistic data term. This is joint work with Daniel Cremers. The final chapter presents a method for high resolution motion layer decomposition and was developed in combination with Daniel Cremers and Thomas Pock. Layer decomposition methods support the notion of a scene model, which allows to model occlusion and enforce temporal consistency. The contributions are twofold: from a practical point of view the proposed method allows to recover fine-detailed layer images by minimizing a single energy. This is achieved by integrating a super-resolution method into the layer decomposition framework. From a theoretical viewpoint the proposed method introduces layer-based regularity terms as well as a graph cut-based scheme to solve for the layer domains. The latter is combined with powerful continuous convex optimization techniques into an alternating minimization scheme. Lastly I want to mention that a significant part of this thesis is devoted to the recent trend of exploiting parallel architectures, in particular graphics cards: many combinatorial algorithms are easily parallelized. In Chapter 3 we will see a case where the standard algorithm is hard to parallelize, but easy for the respective problem instances

Parallel and Flow-Based High Quality Hypergraph Partitioning

Balanced hypergraph partitioning is a classic NP-hard optimization problem that is a fundamental tool in such diverse disciplines as VLSI circuit design, route planning, sharding distributed databases, optimizing communication volume in parallel computing, and accelerating the simulation of quantum circuits. Given a hypergraph and an integer $k$, the task is to divide the vertices into $k$ disjoint blocks with bounded size, while minimizing an objective function on the hyperedges that span multiple blocks. In this dissertation we consider the most commonly used objective, the connectivity metric, where we aim to minimize the number of different blocks connected by each hyperedge. The most successful heuristic for balanced partitioning is the multilevel approach, which consists of three phases. In the coarsening phase, vertex clusters are contracted to obtain a sequence of structurally similar but successively smaller hypergraphs. Once sufficiently small, an initial partition is computed. Lastly, the contractions are successively undone in reverse order, and an iterative improvement algorithm is employed to refine the projected partition on each level. An important aspect in designing practical heuristics for optimization problems is the trade-off between solution quality and running time. The appropriate trade-off depends on the specific application, the size of the data sets, and the computational resources available to solve the problem. Existing algorithms are either slow, sequential and offer high solution quality, or are simple, fast, easy to parallelize, and offer low quality. While this trade-off cannot be avoided entirely, our goal is to close the gaps as much as possible. We achieve this by improving the state of the art in all non-trivial areas of the trade-off landscape with only a few techniques, but employed in two different ways. Furthermore, most research on parallelization has focused on distributed memory, which neglects the greater flexibility of shared-memory algorithms and the wide availability of commodity multi-core machines. In this thesis, we therefore design and revisit fundamental techniques for each phase of the multilevel approach, and develop highly efficient shared-memory parallel implementations thereof. We consider two iterative improvement algorithms, one based on the Fiduccia-Mattheyses (FM) heuristic, and one based on label propagation. For these, we propose a variety of techniques to improve the accuracy of gains when moving vertices in parallel, as well as low-level algorithmic improvements. For coarsening, we present a parallel variant of greedy agglomerative clustering with a novel method to resolve cluster join conflicts on-the-fly. Combined with a preprocessing phase for coarsening based on community detection, a portfolio of from-scratch partitioning algorithms, as well as recursive partitioning with work-stealing, we obtain our first parallel multilevel framework. It is the fastest partitioner known, and achieves medium-high quality, beating all parallel partitioners, and is close to the highest quality sequential partitioner. Our second contribution is a parallelization of an n-level approach, where only one vertex is contracted and uncontracted on each level. This extreme approach aims at high solution quality via very fine-grained, localized refinement, but seems inherently sequential. We devise an asynchronous n-level coarsening scheme based on a hierarchical decomposition of the contractions, as well as a batch-synchronous uncoarsening, and later fully asynchronous uncoarsening. In addition, we adapt our refinement algorithms, and also use the preprocessing and portfolio. This scheme is highly scalable, and achieves the same quality as the highest quality sequential partitioner (which is based on the same components), but is of course slower than our first framework due to fine-grained uncoarsening. The last ingredient for high quality is an iterative improvement algorithm based on maximum flows. In the sequential setting, we first improve an existing idea by solving incremental maximum flow problems, which leads to smaller cuts and is faster due to engineering efforts. Subsequently, we parallelize the maximum flow algorithm and schedule refinements in parallel. Beyond the strive for highest quality, we present a deterministically parallel partitioning framework. We develop deterministic versions of the preprocessing, coarsening, and label propagation refinement. Experimentally, we demonstrate that the penalties for determinism in terms of partition quality and running time are very small. All of our claims are validated through extensive experiments, comparing our algorithms with state-of-the-art solvers on large and diverse benchmark sets. To foster further research, we make our contributions available in our open-source framework Mt-KaHyPar. While it seems inevitable, that with ever increasing problem sizes, we must transition to distributed memory algorithms, the study of shared-memory techniques is not in vain. With the multilevel approach, even the inherently slow techniques have a role to play in fast systems, as they can be employed to boost quality on coarse levels at little expense. Similarly, techniques for shared-memory parallelism are important, both as soon as a coarse graph fits into memory, and as local building blocks in the distributed algorithm