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), 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+) of minimum, where > 0 is a constant.