Max flow vitality in general and $st$-planar graphs

TL;DR

This paper introduces efficient methods to compute the vitality of arcs and nodes in maximum flow problems, achieving significant speedups especially in $st$-planar graphs, with practical implications for network analysis.

Contribution

It presents algorithms to compute the vitality of all arcs and nodes efficiently in general and $st$-planar graphs, reducing computational complexity significantly.

Findings

All arc vitality in general graphs can be computed with 2(n-1) max flow computations.

In $st$-planar graphs, vitality of all arcs and nodes can be computed in O(n) time.

Vitality of contiguous sets of arcs/nodes can be determined efficiently given a planar embedding.

Abstract

The \emph{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.