Searching and sweeping graphs: a brief survey

This papers surveys some of the work done on trying to capture an intruder in a graph. If the intruder may be located only at vertices, the term searching is employed. If the intruder may be located at vertices or along edges, the term sweeping is employed. There are a wide variety of applications for searching and sweeping. Old results, new results and active research directions are discussed

Sweeping Graphs and Digraphs

Searching a network for an intruder is an interesting and difficult problem. Sweeping is one such search model, in which we "sweep" for intruders along edges. The minimum number of sweepers needed to clear a graph G is known as the sweep number sw(G). The sweep number can be restricted by insisting the sweep be monotonic (once an edge is cleared, it must stay cleared) and connected (new clear edges must be incident with already cleared edges). We will examine several lower bounds for sweep number, among them minimum degree, clique number, chromatic number, and girth. We will make use of several of these bounds to calculate sweep numbers for several infinite families of graphs. In particular, these families will answer some open problems regarding the relationships between the monotonic sweep number, connected sweep number, and monotonic connected sweep number. While sweeping originated in simple graphs, the idea may be easily extended to directed graphs, which allow for four different sweep models. We will examine some interesting non-intuitive sweep numbers and look at relations between these models. We also look at bounds on these sweep numbers on digraphs and tournaments

Fast Computation of Supertrees for Compatible Phylogenies with Nested Taxa

Typically, supertree methods combine a collection of source trees in which just the leaves are labeled by taxa. In such methods the resulting supertree is also leaf labeled. An underlying assumption in these methods is that across all trees in the collection, no two of the taxa are nested; for example, "buttercups" and "plants" are nested taxa. Motivated by Page, the first supertree algorithm for allowing the source trees to collectively have nested taxa is called AncestralBuild. Here, in addition to taxa labeling the leaves, the source trees may have taxa labeling some of their interior nodes. Taxa-labeling interior nodes are at a higher taxonomic level than that of their descendants (for example, genera versus species). Analogous to the supertree method Build for deciding the compatibility of a collection of source trees in which just the leaves are labeled, AncestralBuild is a polynomial-time algorithm for deciding the compatibility of a collection of source trees in which some of the interior nodes are also labeled by taxa. Although a more general method, in this paper we show that the original description of AncestralBuild can be modified so that the running time is as fast as the current fastest running time for Build. Fast computation for deciding compatibility is essential if one is to make use of phylogenetic databases that contain thousands of trees on tens of thousands of taxa. This is particularly so as AncestralBuild is incorporated as a basic tool inside more general supertree methods (that is, methods that always output a tree regardless of the compatibility of the source trees). We apply the method to propose a comprehensive phylogeny of the strepsirrhines, a major group of the primates

Project scheduling with modular project completion on a bottleneck resource.

In this paper, we model a research-and-development project as consisting of several modules, with each module containing one or more activities. We examine how to schedule the activities of such a project in order to maximize the expected profit when the activities have a probability of failure and when an activity’s failure can cause its module and thereby the overall project to fail. A module succeeds when at least one of its constituent activities is successfully executed. All activities are scheduled on a scarce resource that is modeled as a single machine. We describe various policy classes, establish the relationship between the classes, develop exact algorithms to optimize over two different classes (one dynamic program and one branch-and-bound algorithm), and examine the computational performance of the algorithms on two randomly generated instance sets.Scheduling; Uncertainty; Research and development; Activity failures; Modular precedence network;

Basic Neutrosophic Algebraic Structures and their Application to Fuzzy and Neutrosophic Models

The involvement of uncertainty of varying degrees when the total of the membership degree exceeds one or less than one, then the newer mathematical paradigm shift, Fuzzy Theory proves appropriate. For the past two or more decades, Fuzzy Theory has become the potent tool to study and analyze uncertainty involved in all problems. But, many real-world problems also abound with the concept of indeterminacy. In this book, the new, powerful tool of neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic models are described. The theory of neutrosophic graphs is introduced and applied to fuzzy and neutrosophic models. This book is organized into four chapters. In Chapter One we introduce some of the basic neutrosophic algebraic structures essential for the further development of the other chapters. Chapter Two recalls basic graph theory definitions and results which has interested us and for which we give the neutrosophic analogues. In this chapter we give the application of graphs in fuzzy models. An entire section is devoted for this purpose. Chapter Three introduces many new neutrosophic concepts in graphs and applies it to the case of neutrosophic cognitive maps and neutrosophic relational maps. The last section of this chapter clearly illustrates how the neutrosophic graphs are utilized in the neutrosophic models. The final chapter gives some problems about neutrosophic graphs which will make one understand this new subject.Comment: 149 pages, 130 figure

Network analysis as applied to a group of AIDS patients linked by sexual contact

Thesis (B.S.) in Psychology -- University of Illinois at Urbana-Champaign, 1989.Includes bibliographical references (leaves 51-56).Microfiche of typescript. [Urbana, Ill.]: Photographic Services, University of Illinois, U of I Library, [1989]. 2 microfiches (83 frames): negative.s 1989 ilu n

The Discrete Acyclic Digraph Markov Model in Data Mining

Graphical Markov models are a powerful tool for the description of complex interactions between the variables of a domain. They provide a succinct description of the joint distribution of the variables. This feature has led to the most successful application of graphical Markov models, that is as the core component of probabilistic expert systems. The fascinating theory behind this type of models arises from three different disciplines, viz., Statistics, Graph Theory and Artificial Intelligence. This interdisciplinary origin has given rich insight from different perspectives. There are two main ways to find the qualitative structure of graphical Markov models. Either the structure is specified by a domain expert or ``structural learning'' is applied, i.e., the structure is automatically recovered from data. For structural learning, one has to compare how well different models describe the data. This is easy for, e.g., acyclic digraph Markov models. However, structural learning is still a hard problem because the number of possible models grows exponentially with the number of variables. The main contributions of this thesis are as follows. Firstly, a new class of graphical Markov models, called TCI models, is introduced. These models can be represented by labeled trees and form the intersection of two previously well-known classes. Secondly, the inclusion order of graphical Markov models is studied. From this study, two new learning algorithms are derived. One for heuristic search and the other for the Markov Chain Monte Carlo Method. Both algorithms improve the results of previous approaches without compromising the computational cost of the learning process. Finally, new diagnostics for convergence assessment of the Markov Chain Monte Carlo Method in structural learning are introduced. The results of this thesis are illustrated using both synthetic and real world datasets

Space-Efficient Algorithms and Verification Schemes for Graph Streams

Structured data-sets are often easy to represent using graphs. The prevalence of massive data-sets in the modern world gives rise to big graphs such as web graphs, social networks, biological networks, and citation graphs. Most of these graphs keep growing continuously and pose two major challenges in their processing: (a) it is infeasible to store them entirely in the memory of a regular server, and (b) even if stored entirely, it is incredibly inefficient to reread the whole graph every time a new query appears. Thus, a natural approach for efficiently processing and analyzing such graphs is reading them as a stream of edge insertions and deletions and maintaining a summary that can be (a) stored in affordable memory (significantly smaller than the input size) and (b) used to detect properties of the original graph. In this thesis, we explore the strengths and limitations of such graph streaming algorithms under three main paradigms: classical or standard streaming, adversarially robust streaming, and streaming verification. In the classical streaming model, an algorithm needs to process an adversarially chosen input stream using space sublinear in the input size and return a desired output at the end of the stream. Here, we study a collection of fundamental directed graph problems like reachability, acyclicity testing, and topological sorting. Our investigation reveals that while most problems are provably hard for general digraphs, they admit efficient algorithms for the special and widely-studied subclass of tournament graphs. Further, we exhibit certain problems that become drastically easier when the stream elements arrive in random order rather than adversarial order, as well as problems that do not get much easier even under this relaxation. Furthermore, we study the graph coloring problem in this model and design color-efficient algorithms using novel parameterizations and establish complexity separations between different versions of the problem. The classical streaming setting assumes that the entire input stream is fixed by an adversary before the algorithm reads it. Many randomized algorithms in this setting, however, fail when the stream is extended by an adaptive adversary based on past outputs received. This is the so-called adversarially robust streaming model. We show that graph coloring is significantly harder in the robust setting than in the classical setting, thus establishing the first such separation for a ``natural\u27\u27 problem. We also design a class of efficient robust coloring algorithms using novel techniques. In classical streaming, many important problems turn out to be ``intractable\u27\u27, i.e., provably impossible to solve in sublinear space. It is then natural to consider an enhanced streaming setting where a space-bounded client outsources the computation to a space-unbounded but untrusted cloud service, who replies with the solution and a supporting ``proof\u27\u27 that the client needs to verify. This is called streaming verification or the annotated streaming model. It allows algorithms or verification schemes for the otherwise intractable problems using both space and proof length sublinear in the input size. We devise efficient schemes that improve upon the state of the art for a variety of fundamental graph problems including triangle counting, maximum matching, topological sorting, maximal independent set, graph connectivity, and shortest paths, as well as for computing frequency-based functions such as distinct items and maximum frequency, which have broad applications in graph streaming. Some of our schemes were conjectured to be impossible, while some others attain smooth and optimal tradeoffs between space and communication costs