Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Network Unreliability in Almost-Linear Time

Ruoxu Cen, Jason Li, Debmalya Panigrahi

TL;DR

This work studies the network unreliability problem $u_G(p)$ for undirected graphs, the probability of disconnection under independent edge failures. The authors design an almost-linear time randomized approximation scheme, achieving $m^{1+o(1)}$ running time, by separating into reliable and unreliable regimes and using Thorup's ideal tree packing to drive efficient importance sampling and a recursive contraction scheme with variance control. The reliable case uses importance sampling guided by ideal tree packing to sample near-minimum cuts efficiently, while the unreliable case uses a carefully tuned recursive contraction (Karger–Stein) and a sparsification-augmented recursion to bound variance; a greedy packing provides approximate loads to realize near-linear time. A bootstrapping step further improves relative bias, and the overall approach nearly closes the linear-time goal for this fundamental problem, with natural extensions and open questions remaining (e.g., dependence, hypergraphs).

Abstract

The network unreliability problem asks for the probability that a given undirected graph gets disconnected when every edge independently fails with a given probability $p$. Valiant (1979) showed that this problem is \#P-hard; therefore, the best we can hope for are approximation algorithms. In a classic result, Karger (1995) obtained the first FPTAS for this problem by leveraging the fact that when a graph disconnects, it almost always does so at a near-minimum cut, and there are only a small (polynomial) number of near-minimum cuts. Since then, a series of results have obtained progressively faster algorithms to the current bound of $m^{1+o(1)} + \tilde{O}(n^{3/2})$ (Cen, He, Li, and Panigrahi, 2024). In this paper, we obtain an $m^{1+o(1)}$-time algorithm for the network unreliability problem. This is essentially optimal, since we need $O(m)$ time to read the input graph. Our main new ingredient is relating network unreliability to an {\em ideal} tree packing of spanning trees (Thorup, 2001).

Network Unreliability in Almost-Linear Time

TL;DR

This work studies the network unreliability problem for undirected graphs, the probability of disconnection under independent edge failures. The authors design an almost-linear time randomized approximation scheme, achieving running time, by separating into reliable and unreliable regimes and using Thorup's ideal tree packing to drive efficient importance sampling and a recursive contraction scheme with variance control. The reliable case uses importance sampling guided by ideal tree packing to sample near-minimum cuts efficiently, while the unreliable case uses a carefully tuned recursive contraction (Karger–Stein) and a sparsification-augmented recursion to bound variance; a greedy packing provides approximate loads to realize near-linear time. A bootstrapping step further improves relative bias, and the overall approach nearly closes the linear-time goal for this fundamental problem, with natural extensions and open questions remaining (e.g., dependence, hypergraphs).

Abstract

The network unreliability problem asks for the probability that a given undirected graph gets disconnected when every edge independently fails with a given probability . Valiant (1979) showed that this problem is \#P-hard; therefore, the best we can hope for are approximation algorithms. In a classic result, Karger (1995) obtained the first FPTAS for this problem by leveraging the fact that when a graph disconnects, it almost always does so at a near-minimum cut, and there are only a small (polynomial) number of near-minimum cuts. Since then, a series of results have obtained progressively faster algorithms to the current bound of (Cen, He, Li, and Panigrahi, 2024). In this paper, we obtain an -time algorithm for the network unreliability problem. This is essentially optimal, since we need time to read the input graph. Our main new ingredient is relating network unreliability to an {\em ideal} tree packing of spanning trees (Thorup, 2001).

Paper Structure

This paper contains 19 sections, 39 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Our Result
  3. Our Techniques
  4. Overall Algorithm and Paper Organization
  5. Preliminaries
  6. The Reliable Case: Importance Sampling on Ideal Tree Packing
  7. Ideal Tree Packing
  8. Surrogate Criterion for Deciding Reliable vs Unreliable Cases
  9. Cut Sampling Algorithm on Tree Packing
  10. Very Reliable Case
  11. Using Bootstrapping to Improve Relative Bias
  12. The Unreliable Case: Recursive Contraction
  13. Description of the Algorithm
  14. Bias of the Estimator
  15. Some Important Properties
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $\varepsilon\in (0, 1)$, there is a randomized Monte Carlo algorithm for the network unreliability problem that runs in $m^{1+o(1)}\varepsilon^{-O(1)}$ time and outputs a $(1\pm\varepsilon)$-approximation to $u_G(p)$ whp.

Theorems & Definitions (77)

  • Theorem 1.1
  • Lemma 1.4
  • proof : Proof of \ref{['thm:main']}
  • Definition 2.1
  • Lemma 2.3: Lemma I.2 of Karger17
  • Definition 2.4
  • Lemma 2.6
  • proof
  • Lemma 3.0
  • Lemma 3.1
  • ...and 67 more