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

Abstract

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

Exploring risk flow attack graph for security risk assessment

“…NF problem has permanently been among the research hotspots in operational research, graph theory and combinatorial optimisation [20]. The solution to this problem generally falls into two categories: the first is to constantly seek a path which can be augmented, and gradually increase the value of flow by adding up the minimum sub-path flow [21,22]; the second is to calculate maximum flow value indirectly by applying the minimum cut theorem [23,24].…”

A recursive formula for the two‐edge connected and biconnected reliability of a complete graph

“…Here we consider the probability that the operational edges of the graph induce a two‐edge connected subgraph and the probability that the operational edges of the graph induce a biconnected, which means two‐vertex connected, subgraph. For recent progress in the two‐terminal case, see . Our main results are recurrence relations for the calculation of and , where denotes the complete graph of order .…”

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“…Indeed effective algorithms are available for numerous network operations to constructively list all minpaths or mincuts, for example [192]. This view leads to a natural inclusion-exclusion strategy [118,189,190,193]. Nevertheless, in an effort to improve upon an exponential time state enumeration method, naive inclusion-exclusion appears to yield an algorithm running in time exponential in the number of minpaths or mincuts, and hence doubly exponential in the number of components.…”

“…As we saw at the outset, structure functions may record much more than operational state, extending reliability. Examples include expected maximum flow value [12,18,80,94,108,118,162,201], expected cost of a flow [40], expected shortest path length [78,90,139,177], shortest path lengths with recourse [11,172], maximum reliability path length [166], minimum cut value [14], expected weight of a minimum spanning tree [27,99,140], maximum weight of a forest [2], diameter [137], expected number of reachable nodes [175], expected cost of a matching [206], and more general performance metrics [70]. Of particular interest is the expected duration of a PERT project [109,110,143,160,194], i.e., the expected length of a longest path in an acyclic, directed graph.…”