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 $c$.

中文翻译：

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多项式时间内的边缘连通性的顶点稀疏化

顶点稀疏化领域中一个重要的开放性问题是是否存在$（1+ \ epsilon）$近似保留切割的顶点稀疏器，其大小接近终端数量。工作Chalermsook等。（SODA 2021）引入了一种称为连通性-$ c $模拟网络的松弛方法，该方法要求构造一个顶点稀疏器，以将$ k $终端之间的连通性保持到$ c $的值，并显示了动态连通性数据结构和可生存的网络设计。我们证明存在具有$ \ widetilde {O}（kc ^ 3）$边缘的连通性$ c $模仿网络，并且可以在多项式时间内以$ n $和$ c $构造，这比Chalermsook等人的结果有所改善。（SODA 2021）对于任何$ c \ ge \ log n $，其运行时间以指数方式取决于$ c $。