Vertex Sparsification for Edge Connectivity in Polynomial Time

An important open question in the area of vertex sparsification is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. The work Chalermsook et al. (SODA 2021) introduced a relaxation called connectivity-$c$ mimicking networks, which asks to construct a vertex sparsifier which preserves connectivity among $k$ terminals exactly up to the value of $c$, and showed applications to dynamic connectivity data structures and survivable network design. We show that connectivity-$c$ mimicking networks with $\widetilde{O}(kc^3)$ edges exist and can be constructed in polynomial time in $n$ and $c$, improving over the results of Chalermsook et al. (SODA 2021) for any $c \ge \log n$, whose runtimes depended exponentially on $c$.