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

Authors:
Harold N. Gabow;Michel X. Goemans;Éva Tardos;David P. Williamson
Affiliations:
University of Colorado at Boulder, Boulder, CO;MIT, Cambridge, MA;Cornell University, Ithaca, NY;Cornell University, Ithaca, NY
Venue:
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2005

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Abstract

The 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. These results prove that the smallest k-ECSS problem gets easier to approximate as k tends to infinity. They also show the integrality gap of the natural linear program is at most 1 + 2/k, for both directed and undirected graphs.