内射瓶颈路径问题的高效在线灵敏度分析
Efficient Online Sensitivity Analysis For The Injective Bottleneck Path Problem
AI总结：
本文研究内射瓶颈路径问题的在线容差分析，提出基于反阿克曼函数的预处理方法，将多源目标对情形下求所有边容差的复杂度从 O(k(m+n log n)) 改进至 O(mα(n,m)+km)。
摘要：
组合优化问题中某个元素关于给定最优解的容差，是指其代价能够减少或增加的最大变化量，使得该解仍然保持最优。瓶颈路径问题是指，给定一个边容量图、一个源点和一个目标点，求源点与目标点之间路径上边容量的最大-最小值。对于该问题的任意给定实例，其具有 n 个顶点和 m 条边，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容差。本文针对预先给定的具有不同边容量的 (n,m)-网络和 k 个源-目标对，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，从而可在 O(k) 时间内求出任意一条边关于成对源点与目标点之间某些最大-最小路径的全部 2k 个容差。为求出所有边关于这些最优路径的两个容差，对于某些 n,m,k，该方法将 Ramaswamy-Orlin-Chakravarty 的复杂度 O(k(m+n log n)) 渐近改进至 O(mα(n,m)+km)。
英文摘要：
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 $O(mα(n,m)+km)$.
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