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 (2`+ 1)-Clique problem in edge weighted graphs requires n2`+1−o(1) time, we prove that for all sparsities of the form m=Θ(n1+1/`), there is no O(n2+mn1−ε) time algorithm for ε > 0 for any of the below problems • Minimum Weight (2`+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. ∗[email protected]. Supported by the EECS Merrill Lynch Fellowship. †[email protected]. Supported by an NSF CAREER Award, NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF Grant BSF:2012338. ‡[email protected]. Supported by an NSF CAREER Award.