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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 problemarXiv:1712.08147v4 fatcat:4kmi3ia3d5cmriclwvyj6oynwa