Vertex Sparsification for Edge Connectivity in Polynomial Time
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An important open question in the area of vertex sparsification is whether (1+ϵ)-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 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 ≥log n, whose runtimes depended exponentially on c.
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