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An FPRAS for two terminal reliability in directed acyclic graphs

Weiming Feng, Heng Guo

TL;DR

A fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs) and shows the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.

Abstract

We give a fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs). In contrast, we also show the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.

An FPRAS for two terminal reliability in directed acyclic graphs

TL;DR

A fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs) and shows the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.

Abstract

We give a fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs). In contrast, we also show the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.
Paper Structure (17 sections, 15 theorems, 100 equations, 1 figure, 4 algorithms)

This paper contains 17 sections, 15 theorems, 100 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm overview
  3. Preliminaries
  4. Problem definitions
  5. More notations
  6. The total variation distance and coupling
  7. The algorithm
  8. The framework of the algorithm
  9. Generate samples
  10. Approximate counting
  11. Analysis
  12. Analysis of Sample
  13. Analysis of ApproxCount
  14. Analyze the main algorithm
  15. #BIS-hardness for s-t unreliability
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be a directed acyclic graph (DAG), failure probabilities $\mathbf{q}=(q_e)_{e \in E} \in [0,1]^E$, two vertices $s,t\in V$, and $\varepsilon>0$. There is a randomized algorithm that takes $(G,\mathbf{q},s,t,\varepsilon)$ as inputs and outputs a $(1\pm\varepsilon)$-approximation to the

Figures (1)

  • Figure 1: An illustration of sampling from $\pi_{u}$

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Remark 4: Crash of Sample
  • Lemma 6
  • proof
  • Corollary 7
  • proof
  • Lemma 8
  • proof
  • ...and 21 more