Consider the following "local" cut-detection problem in a directed graph: We are given a seed vertex x and need to remove at most k edges so that at most ν edges can be reached from x (a "local" cut) or output ⊥ to indicate that no such cut exists. If we are given query access to the input graph, then this problem can in principle be solved without reading the whole graph and with query complexity depending on k and ν. In this paper we consider a slack variant of this problem where, when such a cut exists, we can output a cut with up to O(kν) edges reachable from x.We present a simple randomized algorithm spending O(k 2 ν) time and O(kν) queries for the above variant, improving in particular a previous time bound of O(k O(k) ν) by Chechik et al. [SODA '17]. We also extend our algorithm to handle an approximate variant. We demonstrate that these local algorithms are versatile primitives for designing substantially improved algorithms for classic graph problems by providing the following three applications. (Throughout,Õ(T ) hides polylog(T ).) 1. A randomized algorithm for the classic k-vertex connectivity problem that takes near-linear time when k = O(polylog(n)), namelyÕ(m + nk 3 ) time in undirected graphs. Prior to our work, the state of the art for this range of k were linear-time algorithms for k ≤ 3 [Tarjan FOCS '71; Hopcroft, Tarjan SICOMP '73] and a recent algorithm withÕ(m + n 4/3 k 7/3 ) time [Nanongkai et al. STOC '19]. The story is the same for directed graphs where ourÕ(mk 2 )-time algorithm is near-linear when k = O(polylog(n)). Our techniques also yield an improved approximation scheme. 2. Property testing algorithms for k-edge and -vertex connectivity with query complexities that are near-linear in k, exponentially improving the state-of-the-art. This resolves two open problems, one by Goldreich and Ron [STOC '97] and one by Orenstein and Ron [Theor. Comput. Sci. '11]. 3. A faster algorithm for computing the maximal k-edge connected subgraphs, improving prior work of Chechik et al. [SODA '17]. * This paper resulted from a merge of two papers submitted to arXiv [FY19, NSY19b] and will be presented at the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2020).

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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 higher m, O(m) time is known for k ≤ 3 [Tarjan FOCS'71; Hopcroft, Tarjan SICOMP'73], the first O(n 2 ) time is from [Kanevsky, Ramachandran, FOCS'87] for k = 4 and from [Nagamochi, Ibaraki, Algorithmica'92] for k = O(1). For general k and m, the best bound isÕ(min(kn 2 , n ω + nk ω )) [Henzinger, Rao, Gabow FOCS'96; Linial, Lovász, Wigderson FOCS'86] whereÕ hides polylogarithmic terms and ω < 2.38 is the matrix multiplication exponent.In this paper, we present a randomized Monte Carlo algorithm withÕ(m + k 7/3 n 4/3 ) time for any k = O( √ n). This gives the first subquadratic time bound for any 4 ≤ k ≤ o(n 2/7 ) (subquadratic time refers to O(m) + o(n 2 ) time.) and improves all above classic bounds for all k ≤ n 0.44 . We also present a new randomized Monte Carlo (1 + ǫ)-approximation algorithm that is strictly faster than the previous Henzinger's 2-approximation algorithm [J. Algorithms'97] and all previous exact algorithms. The story is the same for the directed case, where our exactÕ(min{km 2/3 n, km 4/3 })-time for any k = O( √ n) and (1 + ǫ)-approximation algorithms improve classic bounds for small and large k, respectively. Additionally, our algorithm is the first approximation algorithm on directed graphs.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. * Works partially done while at KTH Royal Institute of Technology, Sweden. † Works partially done while at KTH Royal Institute of Technology, Sweden. Theorem 1.4 (Weighted s-t vertex connectivity). Let G = (V, E) be a directed graph with n nodes and m edges where each node has integer weight from [1, U ]. For any s, t ∈ V , in time O(m √ n log n log U )), we can compute deterministically the minimum weight s-t separator S ⊂ V , i.e., s, t / ∈ S and there is no path from s to t in G[V − S]. 3 The conductance of a cut (S, V − S) is defined as Φ(S) = |E(S,V −S)| min{vol(S),vol(V −S)} . 4 They are algorithms based on some random-walk or diffusion process. 5 The algorithms from [KT15, HRW17] in fact do not guarantee non-existence of some low conductance cuts in the second case, but the guarantee is about min-cuts.

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