Subcubic Equivalences Between Path, Matrix, and Triangle Problems

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Subcubic Equivalences Between Path, Matrix, and Triangle Problems

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by Virginia Vassilevska Williams , Ryan Williams

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42 - 11 self

BibTeX

@MISC{Williams_subcubicequivalences,
    author = {Virginia Vassilevska Williams and Ryan Williams},
    title = {Subcubic Equivalences Between Path, Matrix, and Triangle Problems },
    year = {}
}

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Abstract

We say an algorithm on n × n matrices with entries in [−M,M] (or n-node graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The all-pairs shortest paths problem on weighted digraphs (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n 2.99 negative triangles in an edge-weighted graph. • Finding a minimum weight cycle in a graph of non-negative edge weights. • The replacement paths problem on weighted digraphs. • Finding the second shortest simple path between two nodes in a weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min,+)-semiring. Therefore, if APSP cannot be solved in n 3−ε time for any ε> 0, then many other problems also

Keyphrases

subcubic equivalence path triangle problem weighted digraph replacement path problem n-node graph edge-weighted graph non-negative edge weight many important problem many problem negative triangle all-pairs shortest path problem second shortest simple path negative total edge weight weighted problem subcubic reduction minimum weight cycle subcubic reducibility subcubic algorithm weighted graph apsp cannot edge weight matrix product

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