Tight Hardness for Shortest Cycles and Paths in Sparse Graphs.

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 m ≪ n2? 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 = Θ(n1+1/ℓ), there is no O(n2 + mn1−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.