Approximating the smallest k‐edge connected spanning subgraph by LP‐rounding

Abstract

AbstractThe smallest k‐ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k‐edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3/k for undirected graphs with k > 1 odd, 1 + 2/k for undirected graphs with k even, and 1 + 2/k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2/k. On the hardness side we show that for some absolute constant c > 0, for any integer k ≥ 2 (k ≥ 1), a polynomial‐time algorithm approximating the smallest k‐ECSS on undirected (directed) multigraphs to within ratio 1 + c/k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009