Breaking Quadratic Time for Small Vertex Connectivity and an Approximation Scheme
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Vertex connectivity a classic extensively-studied problem. Given an integer k, its goal is to decide if an n-node m-edge graph can be disconnected by removing k vertices. Although a linear-time algorithm was postulated since 1974 [Aho, Hopcroft and Ullman], and despite its sibling problem of edge connectivity being resolved over two decades ago [Karger STOC'96], so far no vertex connectivity algorithms are faster than O(n^2) time even for k=4 and m=O(n). In the simplest case where m=O(n) and k=O(1), the O(n^2) bound dates five decades back to [Kleitman IEEE Trans. Circuit Theory'69]. For general k and m, the best bound is Õ((kn^2, n^ω+nk^ω)). In this paper, we present a randomized Monte Carlo algorithm with Õ(m+k^7/3n^4/3) time for any k=O(√(n)). This gives the first subquadratic time bound for any 4≤ k ≤ o(n^2/7) and improves all above classic bounds for all k< n^0.44. We also present a new randomized Monte Carlo (1+ϵ)-approximation algorithm that is strictly faster than the previous Henzinger's 2-approximation algorithm [J. Algorithms'97] and all previous exact algorithms. The key to our results is to avoid computing single-source connectivity, which was needed by all previous exact algorithms and is not known to admit o(n^2) time. Instead, we design the first local algorithm for computing vertex connectivity; without reading the whole graph, our algorithm can find a separator of size at most k or certify that there is no separator of size at most k `near' a given seed node.
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