Max Flow Vitality of Edges and Vertices in Undirected Planar Graphs

We study the problem of computing the vitality of edges and vertices with respect to $st$-max flow in undirected planar graphs, where the vitality of an edge/vertex in a graph with respect to max flow between two fixed vertices $s,t$ is defined as the max flow decrease when the edge/vertex is removed from the graph. We show that a $delta$ additive approximation of the vitality of all edges with capacity at most $c$ can be computed in $O(rac{c}{delta}n +n log log n)$ time, where $n$ is the size of the graph. A similar result is given for the vitality of vertices. All our algorithms work in $O(n)$ space.