All-pairs bottleneck paths in vertex weighted graphs

Let G = (V,E,w) be a directed graph, where w : V → ℝ is a weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u,v the capacity from u to v, denoted by c(u,v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem the task is to find the capacities for all ordered pairs of vertices. Our main result is an O(n ^2.575) time algorithm for APBP. The exponent is derived from the exponent of fast matrix multiplication. A variant of our algorithm computes shortest paths of maximum bottleneck weight. Let d(u,v) denote the (unweighted) distance from u to v, and let sc(u,v) denote the maximum bottleneck weight of a path from u to v having length d(u,v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u,v) for all ordered pairs of vertices. We present an algorithm for APBSP whose running time is O(n ^2.86).