Tight Conditional Lower Bounds for Vertex Connectivity Problems

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 (Theta) over cap (n(4)), (2) single-source vertex connectivity has complexity (Theta) over cap (n(3)), and (3) single-source-single-sink and global vertex connectivity have complexity (Theta) over cap (n(2)). For graphs with general density, we obtain tight bounds of (Theta) over cap (m(2)), (Theta) over cap (m(1.5)), (Theta) over cap (m), respectively, assuming Gomory-Hu trees for element connectivity can be computed in almost-linear time.