Algebraic methods applied to shortest path and maximum flow problems in stochastic networks

• Published in: Networks Vol. 61; no. 2; pp. 117 - 127
• Main Authors: Hastings, K. C., Shier, D.R.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.03.2013; Wiley; Wiley Subscription Services, Inc
• Subjects: Algebra; algebraic; Algorithms; Applied sciences; Approximation; Arc cutting; Computer science; control theory; systems; Exact sciences and technology; Flows in networks. Combinatorial problems; Information retrieval. Graph; Lower bounds; majorization; maximum flow; minimum cut; Networks; Operational research and scientific management; Operational research. Management science; shortest path; Shortest-path problems; stochastic; Stochasticity; Theoretical computing; algebraic; Probabilistic approach; Network flow; Shortest path; Routing; Source sink relationship; minimum cut; Upper bound; Complex network; Algebraic method; maximum flow; stochastic; majorization; Optimal flow
• ISSN: 0028-3045; 1097-0037
• DOI: 10.1002/net.21485

We present an algebraic approach for computing the distribution of the length of a shortest s ‐ t path, as well as the distribution of the capacity of a minimum s ‐ t cut, in a network where the arc values (lengths and capacities) have known (discrete) probability distributions. For each problem, both exact and approximating algorithms are presented. These approximating algorithms are shown to yield upper and lower bounds on the distribution of interest. This approach likewise provides exact and bounding distributions on the maximum flow value in stochastic networks. We also obtain bounds on the expected shortest path length as well as the expected minimum cut capacity (and the expected maximum flow value). © 2012 Wiley Periodicals, Inc. NETWORKS, 2013