Resolving the Network Reliability Problem with a Tree Decomposition of the Graph

In this paper is given a method to resolve the network reliability problem. This dynamic method need a tree decomposition of the graph, with a small treewidth, and allows to resolve the network reliability problem in linear time. First, the paper explains the tree decomposition and its important properties and then the method is explained with an example of resolution for the all terminal reliability with perfect vertices.

[1]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[2]  M. O. Locks.  Recent Developments in Computing of System-Reliability , 1985, IEEE Transactions on Reliability.

[3]  Michael O. Ball,et al.  Complexity of network reliability computations , 1980, Networks.

[4]  Paul D. Seymour,et al.  Graph minors. I. Excluding a forest , 1983, J. Comb. Theory, Ser. B.

[5]  Nancy G. Kinnersley,et al.  The Vertex Separation Number of a Graph equals its Path-Width , 1992, Inf. Process. Lett..

[6]  Kishor S. Trivedi,et al.  A survey of efficient reliability computation using disjoint products approach , 1995, Networks.

[7]  Corinne Lucet.  Méthode de décomposition pour l'évaluation de la fiabilité des réseaux , 1993 .

[8]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[9]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[10]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[11]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[12]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[13]  Jacques Carlier,et al.  A Decomposition Algorithm for Network Reliability Evaluation , 1996, Discret. Appl. Math..

[14]  Ton Kloks,et al.  Better Algorithms for the Pathwidth and Treewidth of Graphs , 1991, ICALP.

[15]  Olympia Theologou,et al.  Contribution à l'évaluation de la fiabilité des réseaux , 1990 .

[16]  Bruno Courcelle,et al.  Monadic Second-Order Evaluations on Tree-Decomposable Graphs , 1993, Theor. Comput. Sci..

[17]  Hans L. Bodlaender,et al.  Dynamic Programming on Graphs with Bounded Treewidth , 1988, ICALP.

[18]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[19]  J. Carlier,et al.  Factoring and reductions for networks with imperfect vertices , 1991 .

[20]  Bruce A. Reed,et al.  Finding approximate separators and computing tree width quickly , 1992, STOC '92.