Tight Hardness for Shortest Cycles and Paths in Sparse Graphs
Name
1.9781611975031.80.pdf
Description
Published version
Size
849.18 KB
Format
Adobe PDF
Checksum (MD5)
479ac480446643260d106a6532caf64d
Author(s)
Lincoln, Andrea I
Williams, Virginia Vassilevska
Williams, Richard Ryan
Date Issued
January 2018
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/1.9781611975031.80
