Vertex Sparsification for Edge Connectivity in Polynomial Time

Abstract

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 $\tilde{O}(k{c}^3)$ 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 $c$.

16 pages, changed license

References in corpus (2)

• Towards (1+ε)-Approximate Flow Sparsifiers
• On Mimicking Networks Representing Minimum Terminal Cuts