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 $O\Big(m \alpha(m,n)+\min\big((n+k)\log n,km\big)\Big)$-time preprocessing, where $\alpha(\cdot,\cdot)$ 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 $O(m\alpha(n,m)+km)$.

翻译：对于组合优化问题中的一个元素，其关于给定最优解的容忍度是指其成本的最大变化（即减少或增加）范围，使得该解仍保持最优。瓶颈路径问题，给定一个边赋容图、一个源点和一个目标点，旨在寻找源点与目标点之间路径上边容量的最大-最小值。对于具有n个顶点和m条边的该问题的任意给定实例，已知Ramaswamy-Orlin-Chakravarty算法可在O(m+n log n)时间内计算一条最优路径及其所有相关容忍度。本文针对任意预先给定的具有不同边容量的(n,m)网络及k个源-目标点对，提出一种O(m α(m,n) + min((n+k) log n, km))时间的预处理方法，其中α(·,·)为反阿克曼函数，使得可在O(k)时间内计算任意边关于配对源点与目标点之间某些最大-最小路径的2k个容忍度。为计算所有边关于这些最优路径的双向容忍度，该算法在特定n,m,k参数下，将Ramaswamy-Orlin-Chakravarty算法的复杂度O(k(m+n log n))渐进改进至O(m α(n,m) + km)。