How Vulnerable is an Undirected Planar Graph with respect to Max Flow

TL;DR

This paper investigates the vulnerability of undirected planar graphs concerning maximum flow, introducing efficient algorithms to approximate the vitality of edges and vertices with strong theoretical guarantees.

Contribution

It presents new algorithms for approximating edge and vertex vitality in undirected planar graphs, achieving near-optimal efficiency and accuracy, especially for integer capacities.

Findings

High vitality values are well approximated in near $O(n\,\log\log n)$ time.

Algorithms operate in linear space, ensuring efficiency.

Results include improved and sometimes optimal solutions for integer capacities.

Abstract

We study the problem of computing the vitality of edges and vertices with respect to the $st$-max flow in undirected planar graphs, where the vitality of an edge/vertex is the $st$-max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the $st$-max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute $st$-max flow $O(n\log\log n)$. We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in $O(n)$ space.