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

TL;DR

This paper introduces a fast randomized algorithm for local cut detection in directed graphs, enabling near-linear time solutions for connectivity and property testing problems, significantly improving previous bounds and resolving open questions.

Contribution

The paper presents a novel randomized algorithm for local cut detection with improved time and query complexity, and applies it to enhance algorithms for graph connectivity and property testing.

Findings

Achieved near-linear time algorithms for k-vertex connectivity when k=polylogarithmic.

Developed property testing algorithms with near-linear query complexity, resolving open problems.

Improved algorithms for maximal k-edge connected subgraphs.

Abstract

Consider the following "local" cut-detection problem in a directed graph: We are given a seed vertex $x$ and need to remove at most $k$ edges so that at most $\nu$ edges can be reached from $x$ (a "local" cut) or output $\bot$ to indicate that no such cut exists. If we are given query access to the input graph, then this problem can in principle be solved without reading the whole graph and with query complexity depending on $k$ and $\nu$. In this paper we consider a slack variant of this problem where, when such a cut exists, we can output a cut with up to $O(k\nu)$ edges reachable from $x$. We present a simple randomized algorithm spending $O(k^2\nu)$ time and $O(k\nu)$ queries for the above variant, improving in particular a previous time bound of $O(k^{O(k)}\nu)$ by Chechik et al. [SODA '17]. We also extend our algorithm to handle an approximate variant. We demonstrate that these local algorithms are versatile primitives for designing substantially improved algorithms for classic graph problems by providing the following three applications. (Throughout, $\tilde O(T)$ hides $\operatorname{polylog}(T)$.) (1) A randomized algorithm for the classic $k$-vertex connectivity problem that takes near-linear time when $k=O(\operatorname{polylog}(n))$, namely $\tilde O(m+nk^3)$ time in undirected graphs. For directed graphs our $\tilde O(mk^2)$-time algorithm is near-linear when $k=O(\operatorname{polylog}(n))$. Our techniques also yield an improved approximation scheme. (2) Property testing algorithms for $k$-edge and -vertex connectivity with query complexities that are near-linear in $k$, exponentially improving the state-of-the-art. This resolves two open problems, one by Goldreich and Ron [STOC '97] and one by Orenstein and Ron [Theor. Comput Sci. '11]. (3) A faster algorithm for computing the maximal $k$-edge connected subgraphs, improving prior work of Chechik et al. [SODA '17].