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Hypergraph Unreliability in Quasi-Polynomial Time

Ruoxu Cen, Jason Li, Debmalya Panigrahi

TL;DR

This paper gives quasi-polynomial time approximation schemes for the hypergraph unreliability problem and improves the running time to m· nO(log2 n) with an additional exponentially small additive term in the approximation.

Abstract

The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades, and multiple fully polynomial-time approximation schemes are known starting with the work of Karger (STOC 1995). In contrast, prior to this work, no non-trivial result was known for hypergraphs (of arbitrary rank). In this paper, we give quasi-polynomial time approximation schemes for the hypergraph unreliability problem. For any fixed $\varepsilon \in (0, 1)$, we first give a $(1+\varepsilon)$-approximation algorithm that runs in $m^{O(\log n)}$ time on an $m$-hyperedge, $n$-vertex hypergraph. Then, we improve the running time to $m\cdot n^{O(\log^2 n)}$ with an additional exponentially small additive term in the approximation.

Hypergraph Unreliability in Quasi-Polynomial Time

TL;DR

This paper gives quasi-polynomial time approximation schemes for the hypergraph unreliability problem and improves the running time to m· nO(log2 n) with an additional exponentially small additive term in the approximation.

Abstract

The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades, and multiple fully polynomial-time approximation schemes are known starting with the work of Karger (STOC 1995). In contrast, prior to this work, no non-trivial result was known for hypergraphs (of arbitrary rank). In this paper, we give quasi-polynomial time approximation schemes for the hypergraph unreliability problem. For any fixed , we first give a -approximation algorithm that runs in time on an -hyperedge, -vertex hypergraph. Then, we improve the running time to with an additional exponentially small additive term in the approximation.
Paper Structure (41 sections, 31 theorems, 49 equations, 2 figures, 1 algorithm)

This paper contains 41 sections, 31 theorems, 49 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Our Techniques
  4. Organization.
  5. Preliminaries
  6. Hypergraphs.
  7. Random Variables.
  8. Exponential distribution.
  9. Monte Carlo sampling.
  10. DNF probability.
  11. Random Contraction with Large Edge Enumeration
  12. Algorithm Description
  13. Overview.
  14. Base cases.
  15. Recursive case.
  16. ...and 26 more sections

Key Result

Theorem 1.1

For any fixed $\varepsilon\in (0, 1)$, there is a randomized Monte Carlo algorithm for the hypergraph unreliability problem that runs in $m^{O(\log n)}$ time on an $m$-hyperedge, $n$-vertex hypergraph and returns an estimator $X$ that satisfies $X \in (1\pm\varepsilon)u_G(p)$ whp.whp = with high pro

Figures (2)

  • Figure 1: A depiction of a portion of the computation tree. The failed recursive calls are shown in dashed red, while the successful ones are shown in solid black. \ref{['lem:recursive-tree-size-bound']} analyzes the expected size of the recursion tree.
  • Figure 2: A depiction of phases in the computation tree. The filled in nodes are phase nodes. The blue and green nodes respectively root the blue and green phases. Each phase can contain successful recursive steps, shown by solid black edges and black nodes, and failed recursive steps, shown by dashed red edges and red nodes. In a phase, every node has a phase ancestor which is the root node of the phase; for instance, $u$ is the phase ancestor of $v$ (and of every other node in the blue phase).

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3: Capped relative variance
  • Lemma 2.7
  • Lemma 2.13
  • Lemma 2.14
  • Theorem 2.15: karp1989monte
  • ...and 56 more