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