A Faster Local Algorithm for Detecting BoundedSize Cuts with Applications to HigherConnectivity Problems
Abstract
Consider the following "local" cutdetection problem in a directed graph: We are given a starting vertex $s$ and need to detect whether there is a cut with at most $k$ edges crossing the cut such that the side of the cut containing $s$ has at most $\Delta$ interior edges. If we are given query access to the input graph, then this problem can in principle be solved in sublinear time without reading the whole graph and with query complexity depending on $k$ and $\Delta$. We design an elegant randomized procedure that solves a slack variant of this problem with $O(k^2 \Delta)$ queries, improving in particular a previous bound of $O((2(k+1))^{k+2} \Delta)$ by Chechik et al. [SODA 2017]. In this slack variant, the procedure must successfully detect a component containing $s$ with at most $k$ outgoing edges and $\Delta$ interior edges if such a component exists, but the component it actually detects may have up to $O(k \Delta)$ interior edges. Besides being of interest on its own, such cutdetection procedures have been used in many algorithmic applications for higherconnectivity problems. Our new cutdetection procedure therefore almost readily implies (1) a faster vertex connectivity algorithm which in particular has nearly linear running time for polylogarithmic value of the vertex connectivity, (2) a faster algorithm for computing the maximal $k$edge connected subgraphs, and (3) faster property testing algorithms for higher edge and vertex connectivity, which resolves two open problems, one by Goldreich and Ron [STOC '97] and one by Orenstein and Ron [TCS 2011].
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.08382
 Bibcode:
 2019arXiv190408382F
 Keywords:

 Computer Science  Data Structures and Algorithms