Breaking quadratic time for small vertex connectivity and an approximation scheme
Danupon Nanongkai, Thatchaphol Saranurak, Sorrachai Yingchareonthawornchai · 2019
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.