Who's The Weakest Link?

In this paper we consider the following problem: Given a network G, determine if there is an edge in G through which at least c shortest paths pass. This problem arises naturally in various practical situations where there is a massive network (telephone, internet), and routing of data is done via shortest paths and one wants to identify most congested edges in the network.

[1]  David R. Karger Better random sampling algorithms for flows in undirected graphs , 1998, SODA '98.

[2]  Stephen Warshall,et al.  A Theorem on Boolean Matrices , 1962, JACM.

[3]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[4]  David R. Karger,et al.  Random sampling in residual graphs , 2002, STOC '02.

[5]  B. Bollobás The evolution of random graphs , 1984 .

[6]  Gunnar Blom,et al.  Problems and Snapshots from the World of Probability , 1993 .

[7]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[8]  DyerMartin,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991 .

[9]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[10]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[11]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[12]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[13]  Ketan Mulmuley,et al.  Computational geometry - an introduction through randomized algorithms , 1993 .

[14]  Norman L. Johnson,et al.  Urn models and their application , 1977 .

[15]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.

[16]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[17]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[18]  Richard M. Karp,et al.  Monte-Carlo algorithms for the planar multiterminal network reliability problem , 1985, J. Complex..

[19]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[21]  Richard M. Karp,et al.  Monte-Carlo Approximation Algorithms for Enumeration Problems , 1989, J. Algorithms.