Abstract
In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all pairs shortest paths. We present the first truly subcubic algorithm for APBP in general dense graphs. In par- ticular, we give a procedure for computing the (max,min)-product of two arbitrary matrices over R ( {¥, ¥} in O(n 2+w/3 ) O(n 2.792 ) time, where n is the number of vertices and w is the exponent for multiplication over rings. Max-min products can be used to compute the maximum bottleneck values for all pairs of vertices together with a successor matrix from which one can extract an explicit maximum bottleneck path for any pair of vertices in time linear in the length of the path.
Highlights
In recent years, researchers have found surprisingly strong connections between the complexity of fundamental graph problems and the complexity of matrix multiplication over a ring
Several algorithms have been given for solving all pairs shortest paths (APSP) in n3−o(1) time; the most recent development is by Chan [2] and runs in O(n3 log log3 n/ log2 n) time
One is given a directed graph with real edge weights representing capacities, and the problem is to report, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed from s to t along any single path. (This amount is given by the smallest weight edge on the path, a. k. a. the bottleneck edge.) Our algorithm for all pairs bottleneck paths (APBP) runs in O(n2+ω/3) ≤ O(n2.792) time, where ω is the exponent of matrix multiplication over a ring
Summary
Researchers have found surprisingly strong connections between the complexity of fundamental graph problems and the complexity of matrix multiplication over a ring. E., O(n3−δ ) for some constant δ > 0, where n is the number of vertices in the graph Still, it remains to be seen if the general APSP problem can be solved in truly subcubic time. While we are still unable to give a bona fide subcubic algorithm for APSP, we do present such an algorithm for an intimately related problem, all pairs bottleneck paths (APBP). In this problem, one is given a directed graph with (arbitrary) real edge weights representing capacities, and the problem is to report, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed from s to t along any single path. In the algorithms of this paper, the only operations we use on real numbers are comparisons between them
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