Fully Dynamic s-t Edge Connectivity in Subpolynomial Time

We present a deterministic fully dynamic algorithm to answer $c$-edge connectivity queries on pairs of vertices in $n^{o(1)}$ worst case update and query time for any positive integer $c = (\log n)^{o(1)}$ for a graph with $n$ vertices. Previously, only polylogarithmic and $O(\sqrt{n})$ worst case update time fully dynamic algorithms were known for answering $1$, $2$ and $3$-edge connectivity queries respectively [Henzinger and King 1995, Frederikson 1997, Galil and Italiano 1991]. Our result extends the $c$-edge connectivity vertex sparsifier [Chalermsook et al. 2021] to a multi-level sparsification framework. As our main technical contribution, we present a novel update algorithm for the multi-level $c$-edge connectivity vertex sparsifier with subpolynomial update time.