Breaking quadratic time for small vertex connectivity and an approximation scheme

Abstract

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.

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

“…There has been exciting recent work on faster algorithms for vertex connectivity via a local connectivity approach [16,42]. The algorithms are limited to unweighted graphs while our theorem above gives the first constant factor approximation for weighted vertex connectivity in o(mn) time.…”

“…There has been tremendous recent and ongoing progress in fast algorithms for (s, t)-flow and cut problems and leveraging these algorithms for global connectivity is opened up by the new approach. In particular, an important motivating problem is to compute the global (weighted) vertex connectivity of a graph which has received substantial recent attention [16,42]. In this setting we are given a graph G = (V, E) with vertex weights w : V → R + and the goal is to find a minimum weight subset S ⊂ V such that G − S has at least two non-trivial connected components.…”

Computing and Testing Small Connectivity in Near-Linear Time and Queries via Fast Local Cut Algorithms

“…[Kle69, Pod73, ET75, Eve75, Gal80, EH84, Mat87, Bec + 82, LLW88, CT91, NI92, CR94, Hen97, HRG00, Gab06, CGK14]). (See Nanongkai et al [NSY19a] for a more comprehensive literature survey.) For the undirected case, Aho, Hopcroft and Ullman [AHU74,Problem 5.30] asked in their 1974 book for an O(m)-time algorithm for computing κ G , the vertex connectivity of graph G. Prior to our result, O(m)-time algorithms were known only when κ G ≤ 3, due to the classic results of Tarjan [Tar72] and Hopcroft and Tarjan [HT73].…”

“…Table 1 gives an overview of all these results. This result is obtained essentially by using our LocalEC algorithm to solve the local vertex connectivity problem (LocalVC) that was recently studied by Nanongkai, Saranurak, and Yingchareonthawornchai [NSY19a], and then plugging this algorithm into the recent framework of [NSY19a]. The overall algorithm is fairly simple: Let L be one side of the optimal vertex cut M , i.e., M consists of the neighbors of L. We guess the values ν = vol(L) and k = κ, and run our LocalVC algorithm with parameters ν and k on n/ν randomly-selected seed vertices x.…”

“…This problem, which we call LocalEC, has implicitly been studied before in the context of property testing [GR02,OR11] and for developing a centralized algorithm that computes the maximal k-edge connected subgraphs of a directed graph [Che + 17]. Recently, a variant for vertex cuts has been studied for obtaining faster vertex connectivity algorithms [NSY19a]. A somewhat similar problem of locally detecting a small component with low-conductance has recently been studied extensively [KT19,HRW17,SW19], in particular to obtain centralized algorithms for deterministically computing the edge connectivity of an undirected graph.…”