Tight Hardness for Shortest Cycles and Paths in Sparse Graphs
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Fine-grained reductions have established equivalences between many core problems with Õ(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 Õ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn bounds optimal when m ≪ n^2? Starting from the hypothesis that the minimum weight (2ℓ+1)-Clique problem in edge weighted graphs requires n^2ℓ+1-o(1) time, we prove that for all sparsities of the form m = Θ(n^1+1/ℓ), there is no O(n^2 + mn^1-ϵ) time algorithm for ϵ>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.
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