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