Fine-Grained Complexity and Conditional Hardness for Sparse Graphs
  We consider the fine-grained complexity of sparse graph problems that currently have $\tilde{O}(mn)$ time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (minimum weight cycle), and Eccentricities, which is the problem of computing, for each vertex in the graph, the length of a longest shortest path starting at that vertex.   We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the $\tilde{O}(mn)$ class. Surprisingly, very few of the known nontrivial reductions between problems in the $\tilde{O}(mn)$ class are sparse reductions. In the directed case, our results give a partial order on a large collection of problems in the $\tilde{O}(mn)$ class (along with some equivalences). In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to APSP. The latter reduction also gives an improved algorithm for ANSC (for dense graphs).   We propose the MWC Conjecture, a new conditional hardness conjecture that the weight of a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the $\tilde{O}(mn)$ class establish that several problems in this class, including 2-SiSP (second simple shortest path), Radius, and Eccentricities, are MWCC hard. We also identify Eccentricities as a key problem in the $\tilde{O}(mn)$ class which is simultaneously MWCC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than n^k. 
Udit Agarwal and Vijaya Ramachandran
cs.DS cs.CC