The Exponential Time Complexity of Computing the Probability That a Graph is Connected

TL;DR

This paper proves that calculating the probability of a connected graph with random edge failures is computationally exponential in the number of edges, under the Exponential Time Hypothesis.

Contribution

It establishes a tight exponential lower bound on the complexity of computing all-terminal reliability for simple graphs for any fixed edge failure probability.

Findings

Computation time is exponential in Omega(m/ log^2 m) for simple graphs.

The result holds for all fixed probabilities 0 < p < 1.

Supports the conjecture that the problem is inherently hard.

Abstract

We show that for every probability p with 0 < p < 1, computation of all-terminal graph reliability with edge failure probability p requires time exponential in Omega(m/ log^2 m) for simple graphs of m edges under the Exponential Time Hypothesis.