3 research outputs found
Computing and Testing Small Vertex Connectivity in Near-Linear Time and Queries
We present a new, simple, algorithm for the local vertex connectivity problem
(LocalVC) introduced by Nanongkai~et~al. [STOC'19]. Roughly, given an
undirected unweighted graph , a seed vertex , a target volume , and
a target separator size , the goal of LocalVC is to remove vertices
`near' (in terms of ) to disconnect the graph in `local time', which
depends only on parameters and . In this paper, we present a simple
randomized algorithm with running time and correctness probability
.
Plugging our new localVC algorithm in the generic framework of
Nanongkai~et~al. immediately lead to a randomized -time
algorithm for the classic -vertex connectivity problem on undirected graphs.
( hides .) This is the first near-linear time
algorithm for any . Previous fastest algorithm
for small takes time [Nanongkai~et~al.,
STOC'19].
This work is inspired by the algorithm of Chechik~et~al. [SODA'17] for
computing the maximal -edge connected subgraphs. Forster and Yang [arXiv'19]
has independently developed local algorithms similar to ours, and showed that
they lead to an bound for testing -edge and -vertex
connectivity, resolving two long-standing open problems in property testing
since the work of Goldreich and Ron [STOC'97] and Orenstein and Ron [Theor.
Comput. Sci.'11]. Inspired by this, we use local approximation algorithms to
obtain bounds that are near-linear in , namely and
for the bounded and unbounded degree cases,
respectively. For testing -edge connectivity for simple graphs, the bound
can be improved to
Breaking Quadratic Time for Small Vertex Connectivity and an Approximation Scheme
Vertex connectivity a classic extensively-studied problem. Given an integer
, its goal is to decide if an -node -edge graph can be disconnected by
removing 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 time even for and
. In the simplest case where and , the bound
dates five decades back to [Kleitman IEEE Trans. Circuit Theory'69]. For
general and , the best bound is .
In this paper, we present a randomized Monte Carlo algorithm with
time for any . This gives the {\em
first subquadratic time} bound for any and improves
all above classic bounds for all . We also present a new
randomized Monte Carlo -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
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 or certify that there is no separator of size at
most `near' a given seed node
Deterministic Graph Cuts in Subquadratic Time: Sparse, Balanced, and k-Vertex
We study deterministic algorithms for computing graph cuts, with focus on two
fundamental problems: balanced sparse cut and -vertex connectivity for small
(k=O(\polylog n)). Both problems can be solved in near-linear time with
randomized algorithms, but their previous deterministic counterparts take at
least quadratic time. In this paper, we break this bound for both problems.
Interestingly, achieving this for one problem crucially relies on doing so for
the other.
In particular, via a divide-and-conquer argument, a variant of the
cut-matching game by [Khandekar et al.`07], and the local vertex connectivity
algorithm of [Nanongkai et al. STOC'19], we give a subquadratic time algorithm
for -vertex connectivity using a subquadratic time algorithm for computing
balanced sparse cuts on sparse graphs. To achieve the latter, we improve the
previously best bound for approximating balanced sparse cut for the whole
range of . This starts from (1) breaking the barrier on dense graphs
to (where ) using the the PageRank matrix,
but without explicitly sweeping to find sparse cuts; to (2) getting the bound by combining the -trees by [Madry FOCS `10] with the
bound above, and finally; to (3) getting the bound by recursively invoking the second bound in conjunction with
expander-based graph sparsification. Interestingly, our final
bound lands at a natural stopping point in the sense that polynomially breaking
it would lead to a breakthrough for the dynamic connectivity problem.Comment: This manuscript is the merge of several results. Parts of it were
submitted to FOCS'19 and SODA'20. Part of it has since been subsumed by a new
result involving a subset of the authors, arXiv:1910.08025. It's uploaded in
its current form due to its significant technical overlap with the improved
result. We expect to upload splitted, more up to date, versions of this in
the near futur