Fast Computation of Small Cuts via Cycle Space Sampling

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200

Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching

An e#cient heuristic is presented for the problem of finding a minimum-size k- connected spanning subgraph of an (undirected or directed) simple graph G = (V, E). There are four versions of the problem, and the approximation guarantees are as follows:

Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching

An efficient heuristic is presented for the problem of finding a minimum-size k- connected spanning subgraph of an (undirected or directed) simple graph G = (V; E). There are four versions of the problem, and the approximation guarantees are as follows: minimum-size k-node connected spanning subgraph of an undirected graph 1 + [1=k], minimum-size k-node connected spanning subgraph of a directed graph 1 + [1=k], minimum-size k-edge connected spanning subgraph of an undirected graph 1 + [2=(k + 1)], and minimum-size k-edge connected spanning subgraph of a directed graph 1 + [4= p k]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(kjEj 2 ). The following result on simple undirected graphs is used in the analysis: The number of edges required for augmenting a graph of minimum degree k to be k-edge connected is at most k jV j=(k + 1). For undirected graphs and k = 2, a (determi..

Fast Algorithms for k-Shredders and k-Node Connectivity Augmentation

A k-separator (k-shredder) of a k-node connected undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of k-separators is #P-complete. However, we present an O(k 2 n 2 + k 3 n 1:5 )-time (deterministic) algorithm for finding all the k-shredders. This solves an open question: efficiently find a k-separator whose removal maximizes the number of connected components. For k 4, our running time is within a factor of k of the fastest algorithm known for testing k-node connectivity. One application of shredders is in increasing the node connectivity from k to (k + 1) by efficiently adding an (approximately) minimum number of new edges. Jord'an [J. Combinatorial Theory (B) 1995] gave an O(n 5 )-time augmentation algorithm such that the number of new edges is within an additive term of (k \Gamma 2) from a low..

Finding Small Cuts and Building Their Cactus Representation Using Random Circulations (Partial Draft)

It is shown that the recent random circulation algorithm to find cut edges (bridges) and cut pairs can be parallelized optimally. Specifically, the parallel algorithm cam be made to run in O(log n) timeusing O(n + m) operations on the EREW PRAM model where n and m are the number of vertices and edges of the input graph respectively. A second contribution of this paper is an algorithm that runs within the same bounds to build the cactus representation of all cuts of the input graph