24,562 research outputs found

    An Efficient Algorithm for Computing Network Reliability in Small Treewidth

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    We consider the classic problem of Network Reliability. A network is given together with a source vertex, one or more target vertices, and probabilities assigned to each of the edges. Each edge appears in the network with its associated probability and the problem is to determine the probability of having at least one source-to-target path. This problem is known to be NP-hard. We present a linear-time fixed-parameter algorithm based on a parameter called treewidth, which is a measure of tree-likeness of graphs. Network Reliability was already known to be solvable in polynomial time for bounded treewidth, but there were no concrete algorithms and the known methods used complicated structures and were not easy to implement. We provide a significantly simpler and more intuitive algorithm that is much easier to implement. We also report on an implementation of our algorithm and establish the applicability of our approach by providing experimental results on the graphs of subway and transit systems of several major cities, such as London and Tokyo. To the best of our knowledge, this is the first exact algorithm for Network Reliability that can scale to handle real-world instances of the problem.Comment: 14 page

    Exact two-terminal reliability of some directed networks

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    The calculation of network reliability in a probabilistic context has long been an issue of practical and academic importance. Conventional approaches (determination of bounds, sums of disjoint products algorithms, Monte Carlo evaluations, studies of the reliability polynomials, etc.) only provide approximations when the network's size increases, even when nodes do not fail and all edges have the same reliability p. We consider here a directed, generic graph of arbitrary size mimicking real-life long-haul communication networks, and give the exact, analytical solution for the two-terminal reliability. This solution involves a product of transfer matrices, in which individual reliabilities of edges and nodes are taken into account. The special case of identical edge and node reliabilities (p and rho, respectively) is addressed. We consider a case study based on a commonly-used configuration, and assess the influence of the edges being directed (or not) on various measures of network performance. While the two-terminal reliability, the failure frequency and the failure rate of the connection are quite similar, the locations of complex zeros of the two-terminal reliability polynomials exhibit strong differences, and various structure transitions at specific values of rho. The present work could be extended to provide a catalog of exactly solvable networks in terms of reliability, which could be useful as building blocks for new and improved bounds, as well as benchmarks, in the general case

    Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models

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    Recently, we developed and implemented the bond propagation algorithm for calculating the partition function and correlation functions of random bond Ising models in two dimensions. The algorithm is the fastest available for calculating these quantities near the percolation threshold. In this paper, we show how to extend the bond propagation algorithm to directly calculate thermodynamic functions by applying the algorithm to derivatives of the partition function, and we derive explicit expressions for this transformation. We also discuss variations of the original bond propagation procedure within the larger context of Y-Delta-Y-reducibility and discuss the relation of this class of algorithm to other algorithms developed for Ising systems. We conclude with a discussion on the outlook for applying similar algorithms to other models.Comment: 12 pages, 10 figures; submitte

    A contribution to the evaluation and optimization of networks reliability

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    L’évaluation de la fiabilitĂ© des rĂ©seaux est un problĂšme combinatoire trĂšs complexe qui nĂ©cessite des moyens de calcul trĂšs puissants. Plusieurs mĂ©thodes ont Ă©tĂ© proposĂ©es dans la littĂ©rature pour apporter des solutions. Certaines ont Ă©tĂ© programmĂ©es dont notamment les mĂ©thodes d’énumĂ©ration des ensembles minimaux et la factorisation, et d’autres sont restĂ©es Ă  l’état de simples thĂ©ories. Cette thĂšse traite le cas de l’évaluation et l’optimisation de la fiabilitĂ© des rĂ©seaux. Plusieurs problĂšmes ont Ă©tĂ© abordĂ©s dont notamment la mise au point d’une mĂ©thodologie pour la modĂ©lisation des rĂ©seaux en vue de l’évaluation de leur fiabilitĂ©s. Cette mĂ©thodologie a Ă©tĂ© validĂ©e dans le cadre d’un rĂ©seau de radio communication Ă©tendu implantĂ© rĂ©cemment pour couvrir les besoins de toute la province quĂ©bĂ©coise. Plusieurs algorithmes ont aussi Ă©tĂ© Ă©tablis pour gĂ©nĂ©rer les chemins et les coupes minimales pour un rĂ©seau donnĂ©. La gĂ©nĂ©ration des chemins et des coupes constitue une contribution importante dans le processus d’évaluation et d’optimisation de la fiabilitĂ©. Ces algorithmes ont permis de traiter de maniĂšre rapide et efficace plusieurs rĂ©seaux tests ainsi que le rĂ©seau de radio communication provincial. Ils ont Ă©tĂ© par la suite exploitĂ©s pour Ă©valuer la fiabilitĂ© grĂące Ă  une mĂ©thode basĂ©e sur les diagrammes de dĂ©cision binaire. Plusieurs contributions thĂ©oriques ont aussi permis de mettre en place une solution exacte de la fiabilitĂ© des rĂ©seaux stochastiques imparfaits dans le cadre des mĂ©thodes de factorisation. A partir de cette recherche plusieurs outils ont Ă©tĂ© programmĂ©s pour Ă©valuer et optimiser la fiabilitĂ© des rĂ©seaux. Les rĂ©sultats obtenus montrent clairement un gain significatif en temps d’exĂ©cution et en espace de mĂ©moire utilisĂ© par rapport Ă  beaucoup d’autres implĂ©mentations. Mots-clĂ©s: FiabilitĂ©, rĂ©seaux, optimisation, diagrammes de dĂ©cision binaire, ensembles des chemins et coupes minimales, algorithmes, indicateur de Birnbaum, systĂšmes de radio tĂ©lĂ©communication, programmes.Efficient computation of systems reliability is required in many sensitive networks. Despite the increased efficiency of computers and the proliferation of algorithms, the problem of finding good and quickly solutions in the case of large systems remains open. Recently, efficient computation techniques have been recognized as significant advances to solve the problem during a reasonable period of time. However, they are applicable to a special category of networks and more efforts still necessary to generalize a unified method giving exact solution. Assessing the reliability of networks is a very complex combinatorial problem which requires powerful computing resources. Several methods have been proposed in the literature. Some have been implemented including minimal sets enumeration and factoring methods, and others remained as simple theories. This thesis treats the case of networks reliability evaluation and optimization. Several issues were discussed including the development of a methodology for modeling networks and evaluating their reliabilities. This methodology was validated as part of a radio communication network project. In this work, some algorithms have been developed to generate minimal paths and cuts for a given network. The generation of paths and cuts is an important contribution in the process of networks reliability and optimization. These algorithms have been subsequently used to assess reliability by a method based on binary decision diagrams. Several theoretical contributions have been proposed and helped to establish an exact solution of the stochastic networks reliability in which edges and nodes are subject to failure using factoring decomposition theorem. From this research activity, several tools have been implemented and results clearly show a significant gain in time execution and memory space used by comparison to many other implementations. Key-words: Reliability, Networks, optimization, binary decision diagrams, minimal paths set and cuts set, algorithms, Birnbaum performance index, Networks, radio-telecommunication systems, programs

    Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan

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    The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan have been calculated exactly for arbitrary size as well as arbitrary individual edge and node reliabilities, using transfer matrices of dimension four at most. While the all-terminal reliabilities of these graphs are identical, the special case of identical edge (pp) and node (ρ\rho) reliabilities shows that their two-terminal reliabilities are quite distinct, as demonstrated by their generating functions and the locations of the zeros of the reliability polynomials, which undergo structural transitions at ρ=1/2\rho = \displaystyle {1/2}

    Bicriteria Network Design Problems

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    We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur
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