Efficient computation of marginal reliability importance for reducible+ networks in network management

Marginal reliability importance (MRI) of a link is a quantitative measure reflecting the significance of the link in contributing to the terminal-pair reliability (TR) of a given network. The computation of MRI for general networks has been shown to be an NP-complete problem. Moreover, attention has been drawn to a particular set of networks, called reducible networks, which can be simplified to source-sink (two-node) networks via six simple reduction axioms. The computational complexity of the MRI problem for such networks has been shown to be polynomial bounded. In this paper, we first propose a new reduction axiom, referred to as triangle reduction. Networks which can be fully reduced to source-sink networks by the triangle reduction axiom, in addition to the six reduction rules, are further defined as reducible ⁺ networks. For efficient computation of MRI for reducible ⁺ networks, we further propose a two-phase algorithm. The algorithm performs network reduction in the first phase, and backtracks the reduction steps and computes MRIs in the second phase. The two-phase algorithm, as shown, yields a linearly bounded complexity for the computation of MRI for reducible⁺ networks