Efficient Online Sensitivity Analysis For The Injective Bottleneck Path Problem

TL;DR

The paper tackles efficient online sensitivity analysis for the Injective Bottleneck Path Problem by exploiting a maximum spanning tree framework to transform tolerance computations into edge-replacement considerations. It introduces a unified approach using Lowest Common Ancestors and Disjoint-Set structures to enable fast tolerance evaluation across multiple source–target pairs, achieving a preprocessing time of $O(m \alpha(m,n)+\min((n+k)\log n,km))$ and query time of $O(k)$ per edge for $2k$ tolerances. For all edges with respect to the corresponding optimal paths, it achieves $O(m\alpha(m,n)+km)$ total time, representing improvements over the prior $O(k(m+n\log n))$ bound in certain regimes. The methods have practical implications for resilience and rapid reconfiguration in networks under perturbations, enabling scalable sensitivity analysis across multiple demand pairs.

Abstract

The tolerance of an element of a combinatorial optimization problem with respect to a given optimal solution is the maximum change, i.e., decrease or increase, of its cost, such that this solution remains optimal. The bottleneck path problem, for given an edge-capacitated graph, a source, and a target, is to find the $\max$-$\min$ value of edge capacities on paths between the source and the target. For any given sample of this problem with $n$ vertices and $m$ edges, there is known the Ramaswamy-Orlin-Chakravarty's algorithm to compute an optimal path and all tolerances with respect to it in $O(m+n\log n)$ time. In this paper, for any in advance given $(n,m)$-network with distinct edge capacities and $k$ source-target pairs, we propose an $O\Big(m α(m,n)+\min\big((n+k)\log n,km\big)\Big)$-time preprocessing, where $α(\cdot,\cdot)$ is the inverse Ackermann function, to find in $O(k)$ time all $2k$ tolerances of an arbitrary edge with respect to some $\max\min$ paths between the paired sources and targets. To find both tolerances of all edges with respect to those optimal paths, it asymptotically improves, for some $n,m,k$, the Ramaswamy-Orlin-Chakravarty's complexity $O\big(k(m+n\log n)\big)$ up to $O(mα(n,m)+km)$.

Figures (3)

• Figure 1: MSTP upper tolerances replacement edges computation
• Figure 2: MSTP lower tolerances replacement edges computation
• Figure 3: BPP online tolerances computation