Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$ with $m$ edges and $n$ vertices and given an integer $h\geq 1$, outputs in $O(\sqrt{hn}m)$ time such a separator or a $K_h$-minor of $G$. Plotkin, Rao, and Smith gave an $O(hm\sqrt{n\log n})$ time algorithm to find a separator of size $O(h\sqrt{n\log n})$. Kawarabayashi and Reed improved the bound on the size of the separator to $h\sqrt n$ and gave an algorithm that finds such a separator in $O(n^{1 + \epsilon})$ time for any constant $\epsilon > 0$, assuming $h$ is constant. This algorithm has an extremely large dependency on $h$ in the running time (some power tower of $h$ whose height is itself a function of $h$), making it impractical even for small $h$. We are interested in a small polynomial time dependency on $h$ and we show how to find an $O(h\sqrt{n\log n})$-size separator or report that $G$ has a $K_h$-minor in O(\poly(h)n^{5/4 + \epsilon}) time for any constant $\epsilon > 0$. We also present the first O(\poly(h)n) time algorithm to find a separator of size $O(n^c)$ for a constant $c < 1$. As corollaries of our results, we get improved algorithms for shortest paths and maximum matching. Furthermore, for integers $\ell$ and $h$, we give an $O(m + n^{2 + \epsilon}/\ell)$ time algorithm that either produces a $K_h$-minor of depth $O(\ell\log n)$ or a separator of size at most $O(n/\ell + \ell h^2\log n)$. This improves the shallow minor algorithm of Plotkin, Rao, and Smith when $m = \Omega(n^{1 + \epsilon})$. We get a similar running time improvement for an approximation algorithm for the problem of finding a largest $K_h$-minor in a given graph.Comment: To appear at FOCS 201

Computing the block triangular form of a sparse matrix

We consider the problem of permuting the rows and columns of a rectangular or square, unsymmetric sparse matrix to compute its block triangular form. This block triangular form is based on a canonical decomposition of bipartite graphs induced by a maximum matching and was discovered by Dulmage and Mendelsohn. We describe implementations of algorithms to compute the block triangular form and provide computational results on sparse matrices from test collections. Several applications of the block triangular form are also included

High-performance cluster computing, algorithms, implementations and performance evaluation for computation-intensive applications to promote complex scientific research on turbulent flows

Large-scale high-performance computing is a very rapidly growing field of research that plays a vital role in the advance of science, engineering, and modern industrial technology. Increasing sophistication in research has led to a need for bigger and faster computers or computer clusters, and high-performance computer systems are themselves stimulating the redevelopment of the methods of computation. Computing is fast becoming the most frequently used technique to explore new questions. We have developed high-performance computer simulation modeling software system on turbulent flows. Five papers are selected to present here from dozens of papers published in our efforts on complex software system development and knowledge discovery through computer simulations. The first paper describes the end-to-end computer simulation system development and simulation results that help understand the nature of complex shelterbelt turbulent flows. The second paper deals specifically with high-performance algorithm design and implementation in a cluster of computers. The third paper discusses the twelve design processes of parallel algorithms and software system as well as theoretical performance modeling and characterization of cluster computing. The fourth paper is about the computing framework of drag and pressure coefficients. The fifth paper is about simulated evapotranspiration and energy partition of inhomogeneous ecosystems. We discuss the end-to-end computer simulation system software development, distributed parallel computing performance modeling and system performance characterization. We design and compare several parallel implementations of our computer simulation system and show that the performance depends on algorithm design, communication channel pattern, and coding strategies that significantly impact load balancing, speedup, and computing efficiency. For a given cluster communication characteristics and a given problem complexity, there exists an optimal number of nodes. With this computer simulation system, we resolved many historically controversial issues and a lot of important problems

Detección de comunidades en redes: Algoritmos y aplicaciones

El presente trabajo de fin de máster tiene como objetivo la realización de un análisis de los métodos de detección de comunidades en redes. Como parte inicial se realizó un estudio de las características principales de la teoría de grafos y las comunidades, así como medidas comunes en este problema. Posteriormente, se realizó una revisión de los principales métodos de detección de comunidades, elaborando una clasificación, teniendo en cuenta sus características y complejidad computacional, para la detección de las fortalezas y debilidades en los métodos, así como también trabajos posteriores. Luego, se estudio el problema de la calificación de un método de agrupamiento, esto con el fin de evaluar la calidad de las comunidades detectadas, analizando diferentes medidas. Por último se elaboraron las conclusiones así como las posibles líneas de trabajo que se pueden derivar.This master's thesis work has the objective of performing an analysis of the methods for detecting communities in networks. As an initial part, I study of the main features of graph theory and communities, as well as common measures in this problem. Subsequently, I was performed a review of the main methods of detecting communities, developing a classification, taking into account its characteristics and computational complexity for the detection of strengths and weaknesses in the methods, as well as later works. Then, study the problem of classification of a clustering method, this in order to evaluate the quality of the communities detected by analyzing different measures. Finally conclusions are elaborated and possible lines of work that can be derived

A Parallel Graph Partitioning Algorithm for a Message-Passing Multiprocessor

We develop a parallel algorithm for partitioning the vertices of a graph into $p \geq 2$ sets in such a way that few edges connect vertices in different sets. The algorithm is intended for a message-passing multiprocessor system, such as the hypercube, and is based on the Kernighan-Lin algorithm for finding small edge separators on a single processor. We use this parallel partitioning algorithm to find orderings for factoring large sparse symettric positive definite matrices. These orderings not only reduce fill, but also result in good processor utilization and low communication overhead during the factorization. We provide a complexity analysis of the algorithm, as well as some numerical results from an Intel hypercube and a hypercube simulator