24,562 research outputs found
An Efficient Algorithm for Computing Network Reliability in Small Treewidth
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
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
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
LâĂ©valuation 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 dâĂ©numĂ©ration des ensembles minimaux et la factorisation, et dâautres sont restĂ©es Ă lâĂ©tat de simples thĂ©ories. Cette thĂšse traite le cas de lâĂ©valuation et lâoptimisation de la fiabilitĂ© des rĂ©seaux. Plusieurs problĂšmes ont Ă©tĂ© abordĂ©s dont notamment la mise au point dâune mĂ©thodologie pour la modĂ©lisation des rĂ©seaux en vue de lâĂ©valuation de leur fiabilitĂ©s. Cette mĂ©thodologie a Ă©tĂ© validĂ©e dans le cadre dâun 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 dâĂ©valuation et dâoptimisation 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 dâexĂ©cution et en espace de mĂ©moire utilisĂ© par rapport Ă beaucoup dâautres 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
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 () and node ()
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
Bicriteria Network Design Problems
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
- âŠ