Max flow vitality in general and st‐planar graphs

Abstract

The vitality of an arc/node of a graph with respect to the maximum flow between two fixed nodes s and t is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper, we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only 2(n − 1) max flow instances and, in st‐planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in O(n) worst‐case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a “contiguous” set of arcs/nodes in time proportional to the size of the set.

A Linear Time Algorithm for Computing Max-Flow Vitality in Undirected Unweighted Planar Graphs

“…The vitality of an edge e with respect to max-flow measures the max-flow decrement observed when edge e is removed from the graph. A survey on vitality with respect to max-flow problems can be found in [2]. In the same paper, it is proven that:…”

Non-Crossing Shortest Paths in Undirected Unweighted Planar Graphs in Linear Time

“…Multiple pairs shortest paths are also used to compute max-flow in planar undirected graphs [12,13,21]. In particular, they are used to compute the vitality of edges and vertices with respect to the max-flow [1,2].…”

Max-flow vitality in undirected unweighted planar graphs

“…Note that our result also holds for uniform edge weights. In a previous work [2], the same authors presented algorithms for solving the vitality problem for all edges, in the case of general undirected graphs, and for all arcs and nodes, plus contiguous arc sets, for st-planar graphs, both directed and undirected.…”

“…The vitality of an edge e with respect to max-flow measures the max-flow decrement observed when edge e is removed from the graph. A survey on vitality with respect to max-flow problems can be found in [2]. In the same paper, it is shown that:…”