Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

TL;DR

This paper establishes tight conditional lower bounds for several shortest cycle and path problems in sparse graphs, showing that certain algorithms cannot be significantly improved under common complexity hypotheses.

Contribution

It proves that for sparse graphs, many fundamental problems cannot be solved faster than existing algorithms unless certain longstanding complexity hypotheses fail.

Findings

No $O(n^2 + mn^{1- ext{epsilon}})$ algorithms for key problems in sparse graphs.

Conditional lower bounds based on the hardness of the $(2 ext{l}+1)$-Clique problem.

Results extend to unweighted sparse graph problems like $k$-cycle and Wiener index.

Abstract

Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-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 $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the minimum weight $(2\ell+1)$-Clique problem in edge weighted graphs requires $n^{2\ell+1-o(1)}$ time, we prove that for all sparsities of the form $m = \Theta(n^{1+1/\ell})$, there is no $O(n^2 + mn^{1-\epsilon})$ time algorithm for $\epsilon>0$ for \emph{any} of the below problems: Minimum Weight $(2\ell+1)$-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.