Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching (extended abstract)

An efficient heuristic is presented for the problem of finding 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)], and • minimum-size k-edge connected spanning subgraph of a directed graph 1 + [4/ √ k]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple and deterministic and runs in time O(k|E|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 |V |/(k+1). For undirected graphs and k = 2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of (1.5 + ǫ) of minimum, where ǫ > 0 is a constant.