Efficient Online Sensitivity Analysis For The Injective Bottleneck Path Problem

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 this problem and a network with $n$ vertices and $m$ edges, there is known the Ramaswamy-Orlin-Chakravarty's algorithm to compute all tolerances in $O(m+n\log n)$ time. In this paper, for any in advance given sample of the problem with pairwise distinct edge capacities, we present a constant-time algorithm for computing both tolerances of an arbitrary edge with a preprocessing time $O\big(m \alpha(m,n)\big)$, where $\alpha(\cdot,\cdot)$ is the inverse Ackermann function. For given $k$ source-target pairs, our solution yields an $O\big((\alpha(m,n)+k)m\big)$-time algorithm to find tolerances of all edges with respect to optimal paths between the sources and targets, while the known algorithm takes $O\big(k(m+n\log n)\big)$ time to find them.