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Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Author(s)
Lincoln, Andrea I; Williams, Virginia Vassilevska; Williams, Richard Ryan
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Abstract
Fine-grained reductions have established equivalences between many core problems with Õ(n3)-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have Õ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn bounds optimal when mn2? Starting from the hypothesis that the minimum weight (2ℓ+ 1)-Clique problem in edge weighted graphs requires n2ℓ+1-o(1) time, we prove that for all sparsities of the form m = Q(n1+1/ℓ), there is no O(n2 +mn1-ϵ ) time algorithm for e > 0 for any of the below problems Minimum Weight (2ℓ + 1)-Cycle in a directed weighted graph, Shortest Cycle in a directed weighted graph, APSP in a directed or undirected weighted graph, Radius (or Eccentricities) in a directed or undirected weighted graph, Wiener index of a directed or undirected weighted graph, Replacement Paths in a directed weighted graph, Second Shortest Path in a directed weighted graph, Betweenness Centrality of a given node in a directed weighted graph. That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem. Our results also lead to new conditional lower bounds from several related hypothesis for unweighted sparse graph problems including k-cycle, shortest cycle, Radius, Wiener index and APSP.
Date issued
2018-01
URI
https://hdl.handle.net/1721.1/125617
Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Journal
Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Citation
Lincoln, Andrea, Virginia Vassilevska Williams, and Ryan Williams. "Tight Hardness for Shortest Cycles and Paths in Sparse Graphs." ACM-SIAM Symposium on Discrete Algorithms, January 2018, New Orleans, Louisiana, United States, edited by Artur Czumaj, Society for Industrial and Applied Mathematics (SIAM), 2017. © 2020 SIAM
Version: Final published version
ISBN
9781611975031

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