[Paper Review] A polynomial-time approximation algorithm for all-terminal network reliability

Heng Guo, Mark Jerrum|Edinburgh Research Explorer|Sep 25, 2017

Reliability and Maintenance Optimization4 citations

TL;DR

This paper presents the first fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, using a novel analysis of the cluster-popping algorithm in bi-directed graphs. By confirming a conjecture of Gorodezky and Pak, the authors establish that the expected running time of the algorithm is polynomial, enabling efficient approximation of the probability that a network remains connected after random edge failures.

ABSTRACT

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 a undirected graph, assuming each edge fails independently, the remaining graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak (Random Struct. Algorithms, 2014), that the expected running time of the "cluster-popping" algorithm in bi-directed graphs is bounded by a polynomial in the size of the input.

Motivation & Objective

• To develop an efficient approximation algorithm for the all-terminal network reliability problem, which is #P-hard.
• To resolve a long-standing open problem in approximate counting by providing an FPRAS for network reliability.
• To confirm the conjecture by Gorodezky and Pak that the cluster-popping algorithm runs in polynomial expected time on bi-directed graphs.
• To establish a connection between network reliability and reachability in bi-directed graphs, enabling the use of sampling-based approximation.

Proposed method

• The authors employ the cluster-popping algorithm to sample root-connected subgraphs in bi-directed graphs with probability proportional to their weights.
• They use a partial rejection sampling framework to analyze the cluster-popping process and bound the expected running time.
• The method relies on analyzing ratios of subgraph counts $ N_{t-1}/N_t $, showing they are monotonically increasing and bounded within $[1/m, m]$.
• An iterative estimator is constructed using the ratio of observed subgraphs in consecutive size classes to approximate $ N_t $, the number of connected subgraphs of size $ t $.
• The algorithm uses a sequence of contractions and sampling to reduce the problem to estimating edge weight ratios $ r_t = (1-p)/p $, which are estimated via repeated sampling.
• A recursive estimation procedure is applied to compute the total number of connected subgraphs, leading to an FPRAS for the reliability probability.

Experimental results

Research questions

• RQ1 Can the cluster-popping algorithm be shown to run in polynomial expected time on bi-directed graphs, as conjectured by Gorodezky and Pak?
• RQ2 Is there a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem?
• RQ3 Can the equivalence between all-terminal reliability and reachability in bi-directed graphs be leveraged to design an efficient approximation algorithm?
• RQ4 What is the computational complexity of approximating the Tutte polynomial on the line $ x=1, y>1 $, and can it be solved efficiently?

Key findings

• The cluster-popping algorithm runs in polynomial expected time on bi-directed graphs, confirming the conjecture of Gorodezky and Pak.
• An FPRAS is constructed for the all-terminal network reliability problem, providing a fully polynomial-time approximation scheme.
• The expected running time of the sampling algorithm is bounded by a polynomial in the input size, enabling efficient approximation.
• The method achieves accurate estimation of the number of connected subgraphs of each size via iterative ratio estimation.
• The algorithm provides a constant lower bound of $ 1/m $ on the probability of observing subgraphs of size $ t $, ensuring efficient sampling.
• The approach extends to planar graphs, allowing efficient approximation of the Tutte polynomial on the line $ x>1, y=1 $.