We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that in an undirected graph, assuming each edge fails independently, the remainder of the graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak [Random Structures Algorithms, 44 (2014), pp. 201--223] that the expected running time of the “cluster-popping” algorithm in bidirected graphs is bounded by a polynomial in the size of the input.


  1. network reliability
  2. approximate counting
  3. randomized algorithms

MSC codes

  1. 68W20
  2. 68W40
  3. 68Q87

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Information & Authors


Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 964 - 978
ISSN (online): 1095-7111


Submitted: 20 July 2018
Accepted: 23 January 2019
Published online: 9 May 2019


  1. network reliability
  2. approximate counting
  3. randomized algorithms

MSC codes

  1. 68W20
  2. 68W40
  3. 68Q87



Funding Information

Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/N004221/1

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