An e cient heuristic is presented for the problem of nding a minimum-size kconnected 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)],

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... 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 analyses of the heuristics for minimum-size k-node connected spanning subgraphs hinge on theorems of Mader. For undirected graphs and k = 2, a (deterministic) parallel NC version of the heuristic nds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of (1:5 + ) o f m i n i m um, where > 0 is a constant. Introduction Given an undirected or directed simple graph G = ( V E), an e cient approximation algorithm 1 is presented for the problem of nding a k-connected (k = 1 2 3 : : : ) spanning subgraph G 0 = ( V E 0 ) that has the minimum number of edges. Let n and m denote jV j and jEj, respectively. There are four versions of the problem, depending on whether G is a graph (i.e., an undirected graph) or a digraph (i.e., a directed graph), and on whether the spanning subgraph G 0 is required to be k-node connected or k-edge connected. All four versions of the problem are NP-hard: the two problems on graphs are NP-hard for k 2, and the two problems on digraphs are NP-hard for k 1, GJ 79]. 1 An -approximation algorithm for a combinatorial optimization problem runs in polynomial time and delivers a solution whose value is always within the factor of the optimum value. The quantity is called the approximation guarantee of the algorithm.