Efficient processing of hierarchical graphs

The standard representation of a graph is a list of its vertices and edges. However, graphs encountered in some areas have structural regularities that allow them to be represented using considerably less space. Typically, such a succinct encoding consists of a list of basic parts and a set of instructions for assembling the graph from the parts. Several models for succinctly representing graphs and other structures have been studied in the past. These models are capable of representing a graph using space polynomial in the logarithm of the size of the graph. Because of the potentially large difference in size between a graph and its description, it is natural to ask whether there are any problems that can be solved in time polynomial in the size of the succinct description, rather than in the size of the graph. It is known that even simple graph problems become NP-hard or worse under most models for succinct representation. Two exceptions are dynamic graphs and, the model we study, hierarchical graphs;A hierarchical graph [gamma] is a list of graphs and a set of rules describing how to attach the graphs together to form X([gamma]), the graph [gamma] represents. A hierarchical algorithm is an algorithm that, given [gamma], solves some problem defined on X([gamma]). Not every polynomially-solvable graph problem has a polynomial-time hierarchical algorithm. However, polynomial-time hierarchical algorithms for many problems have been developed using a framework called the bottom-up method;We develop a generalization of the bottom-up method that we use to construct and analyze hierarchical algorithms. Our method provides a uniform setting in which to present our algorithms, as well as many developed by other researchers. The problems we study belong to three classes: connectivity augmentation, subgraph homeomorphism, and matroid optimization. The P-connectivity augmentation problem, where P is a connectivity property, is to determine the number of edges that must be added to a graph to satisfy P. We present polynomial-time hierarchical algorithms for bridge-connectivity, biconnectivity, and strong-connectivity augmentation. Series-parallel and outer-planar graphs can be characterized by sets of forbidden graphs closed under homeomorphism. We present linear-time hierarchical algorithms that determine if X([gamma]) is series-parallel or outer-planar, and a polynomial-space hierarchical algorithm that generates a forbidden subgraph of X([gamma]) when one exists. The matroid optimization problem we consider is that of computing costs of optimum bases of matroids defined on graphs. We identify two infinite families of matroids for which polynomial-time hierarchical algorithms for this problem exist. We also develop polynomial-space hierarchical algorithms that generate optimum bases

A Parallel Algorithm for Computing Minimum Spanning Trees

We present a simple and implementable algorithm that computes a minimum spanning tree of an undirected weighted graph G = (V, E) of n = |V| vertices and m = |E| edges on an EREW PRAM in O(log 3=2 n) time using n+m processors. This represents a substantial improvement in the running time over the previous results for this problem using at the same time the weakest of the PRAM models. It also implies the existence of algorithms having the same complexity bounds for the EREW PRAM, for connectivity, ear decomposition, biconnectivity, strong orientation, st-numbering and Euler tours problems

Algorithms for Graph Connectivity and Cut Problems - Connectivity Augmentation, All-Pairs Minimum Cut, and Cut-Based Clustering

We address a collection of related connectivity and cut problems in simple graphs that reach from the augmentation of planar graphs to be k-regular and c-connected to new data structures representing minimum separating cuts and algorithms that smoothly maintain Gomory-Hu trees in evolving graphs, and finally to an analysis of the cut-based clustering approach of Flake et al. and its adaption to dynamic scenarios

An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem

In this thesis, we perform an experimental study of approximation algorithms for the tree augmentation problem (TAP). TAP is a fundamental problem in network design. The goal of TAP is to add the minimum number of edges from a given edge set to a tree so that it becomes 2-edge connected. Formally, given a tree T = (V, E), where V denotes the set of vertices and E denotes the set of edges in the tree, and a set of edges (or links) L ⊆ V × V disjoint from E, the objective is to find a set of edges to add to the tree F ⊆ L such that the augmented tree (V, E ∪ F) is 2-edge connected. Our goal is to establish a baseline performance for each approximation algorithm on actual instances rather than worst-case instances. In particular, we are interested in whether the algorithms rank on practical instances is consistent with their worst-case guarantee rankings. We are also interested in whether preprocessing times, implementation difficulties, and running times justify the use of an algorithm in practice. We profiled and analyzed five approximation algorithms, viz., the Frederickson algorithm, the Nagamochi algorithm, the Even algorithm, the Adjiashivili algorithm, and the Grandoni algorithm. Additionally, we used an integer program and a simple randomized algorithm as benchmarks. The performance of each algorithm was measured using space, time, and quality comparison metrics. We found that the simple randomized is competitive with the approximation algorithms and that the algorithms rank according to their theoretical guarantees. The randomized algorithm is simpler to implement and understand. Furthermore, the randomized algorithm runs faster and uses less space than any of the more sophisticated approximation algorithms

Constrained Planarity and Augmentation Problems

A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex m in T corresponds to a subset of the vertices of the graph called ``cluster''. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown by Dahlhaus, Eades, Feng, Cohen that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In the first part of the thesis, we provide a polynomial time algorithms for c-planarity testing of specific planar clustered graphs: Graphs for which - all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings in T are connected, - for all clusters m G-G(m) is connected. The algorithms are based on the concepts for the subgraph induced planar connectivity augmentation problem, also presented in this thesis. Furthermore, we give some characterizations of c-planar clustered graphs using minors and dual graphs and introduce a c-planar augmentation method. Parts II deals with edge deletion and bimodal crossing minimization. We prove that the maximum planar subgraph problem remains NP-complete even for non-planar graphs without a minor isomorphic to either K(5) or K(3,3), respectively. Further, we investigate the problem of finding a minimum weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Finally, we investigate the problem of drawing a directed graph in two dimensions with a minimal number of crossings such that for every node the incoming and outgoing edges are separated consecutively in the cyclic adjacency lists. It turns out that the planarization method can be adapted such that the number of crossings can be expected to grow only slightly for practical instances

Recent Advances in Fully Dynamic Graph Algorithms

In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms