Breaking quadratic time for small vertex connectivity and an approximation scheme

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Journal Title
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Volume Title
Conference article in proceedings
Date
2019-06-23
Major/Subject
Mcode
Degree programme
Language
en
Pages
12
241-252
Series
STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, Proceedings of the Annual ACM Symposium on Theory of Computing
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(n2) time even for k = 4 and m = O(n). In the simplest case where m = O(n) and k = O(1), the O(n2) 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(n2) 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 is Õ (min(kn2, nω + nkω )) [Henzinger, Rao, Gabow FOCS’96; Linial, Lovász, Wigderson FOCS’86] where Õ hides polylogarithmic terms and ω < 2.38 is the matrix multiplication exponent. In this paper, we present a randomized Monte Carlo algorithm with Õ (m + k7/3n4/3) time for any k = O(n). This gives the first subquadratic time bound for any 4 ≤ k ≤ o(n2/7) (subquadratic time refers to O(m) + o(n2) time.) and improves all above classic bounds for all k ≤ n0.44. 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 story is the same for the directed case, where our exact Õ (min(km2/3n, km4/3))-time for any k = O(n) and (1 + ϵ)-approximation algorithms improve all previous exact bounds. 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(n2) 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.
Description
| openaire: EC/H2020/715672/EU//DisDyn
Keywords
Graph algorithms, Local flow algorithms, Vertex connectivity
Other note
Citation
Nanongkai , D , Saranurak , T & Yingchareonthawornchai , S 2019 , Breaking quadratic time for small vertex connectivity and an approximation scheme . in M Charikar & E Cohen (eds) , STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing . Proceedings of the Annual ACM Symposium on Theory of Computing , ACM , pp. 241-252 , ACM Symposium on Theory of Computing , Phoenix , Arizona , United States , 23/06/2019 . https://doi.org/10.1145/3313276.3316394