Estimation of Small s-t Reliabilities in Acyclic Networks

TL;DR

This paper presents an improved Monte-Carlo method for estimating small s-t reliabilities in directed acyclic networks, offering sharper bounds and variance reduction, and demonstrates its effectiveness on large-scale networks.

Contribution

It introduces a specialized Monte-Carlo algorithm with variance reduction for efficiently approximating small s-t reliabilities in acyclic networks, improving bounds and scalability.

Findings

Sharper worst-case bounds for sample size in uniform failure probability case

Variance reduction technique decreases the number of iterations needed

Method effectively estimates small reliabilities on networks with up to one million vertices

Abstract

In the classical s-t network reliability problem a fixed network G is given including two designated vertices s and t (called terminals). The edges are subject to independent random failure, and the task is to compute the probability that s and t are connected in the resulting network, which is known to be #P-complete. In this paper we are interested in approximating the s-t reliability in case of a directed acyclic original network G. We introduce and analyze a specialized version of the Monte-Carlo algorithm given by Karp and Luby. For the case of uniform edge failure probabilities, we give a worst-case bound on the number of samples that have to be drawn to obtain an epsilon-delta approximation, being sharper than the original upper bound. We also derive a variance reduction of the estimator which reduces the expected number of iterations to perform to achieve the desired accuracy when applied in conjunction with different stopping rules. Initial computational results on two types of random networks (directed acyclic Delaunay graphs and a slightly modified version of a classical random graph) with up to one million vertices are presented. These results show the advantage of the introduced Monte-Carlo approach compared to direct simulation when small reliabilities have to be estimated and demonstrate its applicability on large-scale instances.