Algorithms, parallelism and fine-grained complexity for shortest path problems in sparse graphs [article]

Udit Agarwal, Austin, The University Of Texas At, Vijaya Ramachandran
Computation of shortest paths is one of the classical problems in theoretical computer science. Given a pair of nodes s and t in a graph G, the goal is to find a path of minimum weight from s to t. Most graphs that commonly occur in practice are sparse graphs. In this work, we deal with several computational problems related to shortest paths in sparse graphs and we present algorithms that provide significant improvements in performance in both sequential and distributed settings. We also
more » ... t fine-grained reductions that establish fine-grained hardness for several problems related to shortest paths. In the sequential context, we consider the fine-grained complexity of sparse graph problems whose time complexities have stayed at Õ(mn) over the past several decades, where m is the number of edges and n is the number of vertices in the input graph. All of these problems are known to be subcubic equivalent and this shows that achieving sub-mn running time is hard, but only for dense graphs where $m = [Theta] (n²). 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 Õ(mn) class. We also introduce the MWC-hardness conjecture, which states that Minimum Weight Cycle problem cannot be solved in sub-mn time. We establish that several important graph problems in the Õ(mn) class such as APSP, second simple shortest path (2-SiSP), Radius, and Betweenness Centrality are MWC-Hard, establishing sub-mn fine-grained hardness for these problems. A well-known generalization of the shortest path problem is the k-simple shortest paths (k-SiSP) problem, where we want to find k simple paths from s to t in a non-decreasing order of their weight. In this thesis we present a new approach for computing all pairs k simple shortest paths (k-APSiSP), which is based on forming suitable path extensions to find simple shortest paths; this method is different from the 'detour finding' technique used in all prior wo [...]
doi:10.26153/tsw/10160 fatcat:sgpa6t43rbfsbkhrb7gmccenmq