# Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

@article{Lincoln2018TightHF,
title={Tight Hardness for Shortest Cycles and Paths in Sparse Graphs},
author={Andrea Lincoln and Virginia Vassilevska Williams and Richard Ryan Williams},
journal={ArXiv},
year={2018},
volume={abs/1712.08147}
}
• Published 21 December 2017
• Computer Science, Mathematics
• ArXiv
Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-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 $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the…

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