Computing and Testing Small Vertex Connectivity in Near-Linear Time and Queries
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We present a new, simple, algorithm for the local vertex connectivity problem (LocalVC) introduced by Nanongkai et al. [STOC'19]. Roughly, given an undirected unweighted graph G, a seed vertex x, a target volume ν, and a target separator size k, the goal of LocalVC is to remove k vertices `near' x (in terms of ν) to disconnect the graph in `local time', which depends only on parameters ν and k. In this paper, we present a simple randomized algorithm with running time O(ν k^2) and correctness probability 2/3. Plugging our new localVC algorithm in the generic framework of Nanongkai et al. immediately lead to a randomized Õ(m+nk^3)-time algorithm for the classic k-vertex connectivity problem on undirected graphs. (Õ(T) hides polylog(T).) This is the first near-linear time algorithm for any 4≤ k ≤polylog n. Previous fastest algorithm for small k takes Õ(m+n^4/3k^7/3) time [Nanongkai et al., STOC'19]. This work is inspired by the algorithm of Chechik et al. [SODA'17] for computing the maximal k-edge connected subgraphs. Forster and Yang [arXiv'19] has independently developed local algorithms similar to ours, and showed that they lead to an Õ(k^3/ϵ) bound for testing k-edge and -vertex connectivity, resolving two long-standing open problems in property testing since the work of Goldreich and Ron [STOC'97] and Orenstein and Ron [Theor. Comput. Sci.'11]. Inspired by this, we use local approximation algorithms to obtain bounds that are near-linear in k, namely Õ(k/ϵ) and Õ(k/ϵ^2) for the bounded and unbounded degree cases, respectively. For testing k-edge connectivity for simple graphs, the bound can be improved to Õ((k/ϵ, 1/ϵ^2)).
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