Diameter Constrained Reliability: Computational Complexity in terms of the diameter and number of terminals

TL;DR

This paper investigates the computational complexity of diameter-constrained reliability in graphs, revealing polynomial-time solvability for small diameters or fixed terminal counts, and NP-hardness in more general cases, including new cases where all nodes are terminals.

Contribution

The paper establishes the NP-hardness of diameter-constrained reliability when all nodes are terminals and diameter is at least 2, filling a gap in the existing literature.

Findings

DCR is polynomial-time solvable for diameter 1 or diameter 2 with fixed terminals.

DCR is NP-hard when the number of terminals is at least 2 and diameter is 3 or more.

The NP-hardness of the case with all nodes as terminals and diameter ≥ 2 is proven for the first time.

Abstract

Let $G=(V,E)$ be a simple graph with $|V|=n$ nodes and $|E|=m$ links, a subset $K \subseteq V$ of \emph{terminals}, a vector $p=(p_1,\ldots,p_m) \in [0,1]^m$ and a positive integer $d$, called \emph{diameter}. We assume nodes are perfect but links fail stochastically and independently, with probabilities $q_i=1-p_i$. The \emph{diameter-constrained reliability} (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by $d$ links, or less. This number is denoted by $R_{K,G}^{d}(p)$. The general DCR computation is inside the class of $\mathcal{N}\mathcal{P}$-Hard problems, since is subsumes the complexity that a random graph is connected. In this paper, the computational complexity of DCR-subproblems is discussed in terms of the number of terminal nodes $k=|K|$ and diameter $d$. Either when $d=1$ or when $d=2$ and $k$ is fixed, the DCR is inside the class $\mathcal{P}$ of polynomial-time problems. The DCR turns $\mathcal{N}\mathcal{P}$-Hard when $k \geq 2$ is a fixed input parameter and $d\geq 3$. The case where $k=n$ and $d \geq 2$ is fixed are not studied in prior literature. Here, the $\mathcal{N}\mathcal{P}$-Hardness of this case is established.