Very simple methods for all pairs network flow analysis
SIAM Journal on Computing
Random sampling in cut, flow, and network design problems
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
A matroid approach to finding edge connectivity and packing arborescences
Selected papers of the 23rd annual ACM symposium on Theory of computing
Preserving and Increasing Local Edge-Connectivity in Mixed Graphs
SIAM Journal on Discrete Mathematics
Beyond the flow decomposition barrier
Journal of the ACM (JACM)
Augmenting undirected edge connectivity in Õ(n2) time
Journal of Algorithms
A fast algorithm for computing steiner edge connectivity
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A simple minimum T-cut algorithm
Discrete Applied Mathematics
An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Efficient edge splitting-off algorithms maintaining all-pairs edge-connectivities
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Finding maximal k-edge-connected subgraphs from a large graph
Proceedings of the 15th International Conference on Extending Database Technology
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Given an undirected unweighted graph G = (V, E) and an integer k ≥ 1, we consider the problem of computing the edge connectivities of all those (s, t) vertex pairs, whose edge connectivity is at most k. We present an algorithm with expected running time Õ(m + nk3) for this problem, where |V| = n and |E| = m. Our output is a weighted tree T whose nodes are the sets V1, V2,..., V l of a partition of V, with the property that the edge connectivity in G between any two vertices s ε Vi and t ε Vj, for i ≠ j, is equal to the weight of the lightest edge on the path between Vi and Vj in T. Also, two vertices s and t belong to the same Vi for any i if and only if they have an edge connectivity greater than k. Currently, the best algorithm for this problem needs to compute all-pairs min-cuts in an O(nk) edge graph; this takes Õ(m + n5/2kmin{k1/2, n1/6}) time. Our algorithm is much faster for small values of k; in fact, it is faster whenever k is o(n5/6). Our algorithm yields the useful corollary that in Õ(m + nc3) time, where c is the size of the global min-cut, we can compute the edge connectivities of all those pairs of vertices whose edge connectivity is at most αc for some constant α. We also present an Õ(m + n) Monte Carlo algorithm for the approximate version of this problem. This algorithm is applicable to weighted graphs as well. Our algorithm, with some modifications, also solves another problem called the minimum T-cut problem. Given T ⊆ V of even cardinality, we present an Õ(m + nk3) algorithm to compute a minimum cut that splits T into two odd cardinality components, where k is the size of this cut.