A Fast Algorithm for Searching All Multi-State Minimal Cuts

Abstract

Evaluating network reliability is an important topic in planning, designing, and control of systems. Many real-world systems are limited-flow (multi-state) networks composed of multi-state components, and their reliabilities can be computed in terms of minimal cut (MC) vectors to level <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> (named <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC). A straightforward exact algorithm is presented here. The idea is to find all <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC prior to calculating the limited-flow network reliability between the source, and the sink nodes (i.e. the one-to-one reliability), under the condition that all MC are known in advance. The proposed method is more efficient and effective compared to the existing algorithms in deleting the duplicate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC, verifying <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC candidates, and finding the lower bounds of edge capacities in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC. The concepts of the proposed algorithm are novel, and first developed, followed by the analysis of associated computational complexities. Further, an example is provided to illustrate the generation of all <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC, and verify that the obtained <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -MC are not duplicated.