All-Pairs Bottleneck Paths For General Graphs in Truly Sub-Cubic Time

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All-Pairs Bottleneck Paths For General Graphs in Truly Sub-Cubic Time (2007)

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by Virginia Vassilevska , Ryan Williams , Raphael Yuster

Venue:	STOC'07
Citations:	12 - 6 self

BibTeX

@MISC{Vassilevska07all-pairsbottleneck,
    author = {Virginia Vassilevska and Ryan Williams and Raphael Yuster},
    title = {All-Pairs Bottleneck Paths For General Graphs in Truly Sub-Cubic Time },
    year = {2007}
}

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Abstract

In the all-pairs bottleneck paths (APBP) problem (a.k.a. allpairs maximum capacity paths), one is given a directed graph with real non-negative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can be routed 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 sub-cubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max,min)-product of two arbitrary matrices over R ∪ {∞, −∞} in O(n 2+ω/3) ≤ O(n 2.792) time, where n is the number of vertices and ω is the exponent for matrix multiplication over rings. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length of the path.

Keyphrases

all-pairs bottleneck path general graph truly sub-cubic time general dense graph apbp problem allpairs maximum capacity path explicit maximum bottleneck path arbitrary matrix directed graph maximum flow all-pairs shortest path operation research real non-negative capacity time linear single path maximum amount matrix multiplication sub-cubic algorithm

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