Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets. >

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Petri nets: Properties, analysis and applications

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The field of metaheuristics for the application to combinatorial optimization problems is a rapidly growing field of research. This is due to the importance of combinatorial optimization problems for the scientific as well as the industrial world. We give a survey of the nowadays most important metaheuristics from a conceptual point of view. We outline the different components and concepts that are used in the different metaheuristics in order to analyze their similarities and differences. Two very important concepts in metaheuristics are intensification and diversification. These are the two forces that largely determine the behavior of a metaheuristic. They are in some way contrary but also complementary to each other. We introduce a framework, that we call the I&D frame, in order to put different intensification and diversification components into relation with each other. Outlining the advantages and disadvantages of different metaheuristic approaches we conclude by pointing out the importance of hybridization of metaheuristics as well as the integration of metaheuristics and other methods for optimization.

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Metaheuristics in combinatorial optimization: Overview and conceptual comparison

1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑

Network Flows

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Basic Concepts. Trees, Cutsets, and Circuits. Eulerian and Hamiltonian Graphs. Graphs and Vector Spaces. Directed Graphs. Matrices of a Graph. Planarity and Duality. Connectivity and Matching. Covering and Coloring. Matroids. Graph Algorithms. Flows in Networks. Indexes.

Graphs: Theory and Algorithms

Graphs, networks, and algorithms

We study the multi-constrained quality-of-service (QoS) routing problem where one seeks to find a path from a source to a destination in the presence of K ges 2 additive end-to-end QoS constraints. This problem is NP-hard and is commonly modeled using a graph with n vertices and m edges with K additive QoS parameters associated with each edge. For the case of K = 2, the problem has been well studied, with several provably good polynomial time-approximation algorithms reported in the literature, which enforce one constraint while approximating the other. We first focus on an optimization version of the problem where we enforce the first constraint and approximate the other K - 1 constraints. We present an O(mn log log log n + mn/epsi) time (1 + epsi)(K - 1)-approximation algorithm and an O(mn log log log n + m(n/epsi)K-1) time (1 + epsi)-approximation algorithm, for any epsi > 0. When K is reduced to 2, both algorithms produce an (1 + epsi)-approximation with a time complexity better than that of the best-known algorithm designed for this special case. We then study the decision version of the problem and present an O(m(n/epsi)K-1) time algorithm which either finds a feasible solution or confirms that there does not exist a source-destination path whose first weight is bounded by the first constraint and whose every other weight is bounded by (1 - epsi) times the corresponding constraint. If there exists an H-hop source-destination path whose first weight is bounded by the first constraint and whose every other weight is bounded by (1 - epsi) times the corresponding constraint, our algorithm finds a feasible path in O(m(H/epsi)K-1) time. This algorithm improves previous best-known algorithms with O((m + n log n)n/epsi) time for K = 2 and 0(mn(n/epsi)K-1) time for if ges 2.

Polynomial time approximation algorithms for multi-constrained QoS routing

We study quality-of-service (QoS) and quality-of-protection (QoP) issues in redundant tree based preplanned recovery schemes for a single-link failure in two-edge connected graphs and for a single-node failure in two-connected graphs. We present schemes (to be called G-MFBG schemes) that generalize the schemes (to be called MFBG schemes) developed by Me/spl acute/dard et al. (1997) to construct a pair of redundant trees, called red and blue trees, which guarantees fast recovery from any single-link/node failure, as long as the failed node is not the root node. Using the G-MFBG schemes, we study QoS issues relating to red/blue trees. We present effective heuristics for computing a pair of redundant trees with low average delay or small total cost. We develop an optimal algorithm for computing a pair of red/blue trees with maximum bandwidth. Furthermore, a pair of red/blue trees guarantees fast recovery from simultaneous multiple failures if it satisfies certain properties. This leads us to define the concept of QoP of a pair of red/blue trees. We present an effective heuristic to construct a pair of red/blue trees with high QoP. The paper concludes with a discussion of computational results that demonstrate the effectiveness of the different algorithms presented.

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Quality-of-service and quality-of-protection issues in preplanned recovery schemes using redundant trees

The authors present two O(n/sup 2/) planarization algorithms, PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on A. Lempel, S. Even, and I. Cederbaum's (1967) planarity testing algorithm and its implementation using PQ-trees. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. The algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G/sub p/ of a nonplanar graph G, the MAXIMAL-PLANARIZE algorithm constructs a maximal planar subgraph of G which contains G/sub p/. This latter algorithm can also be used to planarize maximally a biconnected planar graph. >

Krishnaiyan Thulasiraman

Papers

Graphs: Theory and Algorithms

Graphs, networks, and algorithms

Polynomial time approximation algorithms for multi-constrained QoS routing

Quality-of-service and quality-of-protection issues in preplanned recovery schemes using redundant trees

O(n/sup 2/) algorithms for graph planarization