We study the fine-grained complexity of graph connectivity problems in unweighted undirected graphs. Recent development shows that all variants of edge connectivity problems, including single-source-single-sink, global, Steiner, single-source, and all-pairs connectivity, are solvable in $m^{ 1+o(1)}$ time, collapsing the complexity of these problems into the almost-linear-time regime. While, historically, vertex connectivity has been much harder, the recent results showed that both single-source-single-sink and global vertex connectivity can be solved in $m^{ 1+o(1)}$ time, raising the hope of putting all variants of vertex connectivity problems into the almost-linear-time regime too. We show that this hope is impossible, assuming conjectures on finding 4-cliques. Moreover, we essentially settle the complexity landscape by giving tight bounds for combinatorial algorithms in dense graphs. There are three separate regimes: (1) all-pairs and Steiner vertex connectivity have complexity $\hat{ \Theta}(n^{ 4})$, (2) single-source vertex connectivity has complexity $\hat{ \Theta}(n^{ 3})$, and (3) single-source-single-sink and global vertex connectivity have complexity $\hat{ \Theta}(n^{ 2})$. For graphs with general density, we obtain tight bounds of $\hat{ \Theta}(m^{ 2})$, $\hat{ \Theta}(m^{ 1.5})$, $\hat{ \Theta}(m)$, respectively, assuming Gomory-Hu trees for element connectivity can be computed in almost-linear time.
Categories: cs.DS