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Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

Vertex Sparsification for Edge Connectivity

Abstract

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether (1 + )-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter c, find a smaller graph, which we call connectivity-c mimicking network, which preserves connectivity among k terminals exactly up to the value of c. We show that connectivity-c mimicking networks with O(kc4) edges exist and can be found in time m(c log n)O(c). We also give a separate algorithm that constructs such graphs with k · O(c)2c edges in time mcO(c) logO(1) n.
These results lead to the first data structures for answering fully dynamic offline c-edge-connectivity queries for c ≥ 4 in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

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cover image Proceedings
Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 1206 - 1225
Editor: Dániel Marx, CISPA Helmholtz Center for Information Security, Germany
ISBN (Online): 978-1-61197-646-5

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Published online: 7 January 2021

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