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 a given (undirected or directed) graph G=(V,E). There are four versions of the problem, depending on whether G is undirected or directed, and whether the spanning subgraph is required to be k-node connected (k-NCSS) or k-edge connected (k-ECSS). The approximation guarantees are as follows: min-size k-NCSS of an undirected graph 1+[1/k], min-size k-NCSS of a directed graph 1+[1/k], min-size k-ECSS of an undirected graph 1+[7/k], & min-size k-ECSS 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, deterministic, and runs in time O(k|E|²). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or a-edge connected) spanning subgraph whose size is within a factor of (1.5+ε) of minimum, where ε>0 is a constant