Subcubic Equivalences between Path, Matrix and Triangle Problems
By Virginia Vassilevska Williams, Ryan Williams. IEEE Annual Symposium on Foundations of Computer Science, 2010
488 citations recorded by the source corpus.
We say an algorithm on n by n matrices with entries in [-M, M] (or n-node graphs with edge weights from [-M, M]) is truly sub cubic if it runs in O(n^{3-\delta} \poly(\log M)) time for some \delta > 0. We define a notion of sub cubic reducibility, and show that many important problems on graphs and matrices solvable in O(n^3) time are equivalent under sub cubic reductions. Namely, the following weighted problems either all have truly sub cubic algorithms, or none of them do: - The all-pairs shortest paths problem (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 in an edge-weighted digraph. - Finding the second shortest simple path between two nodes in an edge-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-\eps} time for any \eps > 0, then many other problems also need essentially cubic time. In fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure. These equivalences simplify prior work on sub cubic algorithms for all-pairs path problems, since it now suffices to give appropriate sub cubic triangle detection algorithms. Other consequences of our work are new combinatorial approaches to Boolean matrix multiplication over the (OR, AND)-semiring (abbreviated as BMM). We show that practical advances in triangle detection would imply practical BMM algorithms, among other results. Building on our techniques, we give two new BMM algorithms: a derandomization of the recent combinatorial BMM algorithm of Bansal and Williams (FOCS'09), and an improved quantum algorithm for BMM.
Subfields: Computational Theory and Mathematics, Computer Networks and Communications
View Paper: https://www.semanticscholar.org/paper/361dcfc389dbe51c94957fe7eb126055449a174e
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In Computational Theory and Mathematics
- Amir Abboud
- Oren Weimann
- Virginia Vassilevska Williams
- F. Gall
- James B. Orlin
- Daniel A. Spielman
- Shang-Hua Teng
In Computer Networks and Communications
Citations in the corpus
- Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems (2014, methodology): “Roditty and Vassilevska Williams showed that any (3/2−ε)-approximation algorithm that runs in O(n2−δ) time in undirected unweighted graphs with Õ(n) edges, for any constants ε, δ > 0 would violate the SETH.”
- Consequences of Faster Alignment of Sequences (2014, background): “It is a longstanding open problem whether APSP on n-node graphs can be solved in O(n3−ε) time [58].”