Better performance bounds for finding the smallest k-edge connected spanning subgraph of a multigraph

• Authors: Harold N. Gabow
• Venue: SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2003

Khuller and Raghavachari [12] present an approximation algorithm (the KR algorithm) for finding the smallest k-edge connected spanning subgraph (k-ECSS) of an undirected multigraph. They prove the KR algorithm has approximation ratio e k this requires a minor modification of the algorithm. This is the bestknown performance bound for the smallest k-ECSS problem for arbitrary k. Our analysis also gives the best-known performance bound for any fixed value of k ≤ 3, e.g., for even k the approximation ratio is ≤ 1 + (1 -- 1/k)k/2. Our analysis is based on a laminar family of sets (similar to families used in related contexts) which gives a better accounting of edges added in previous iterations of the algorithm. We also present a polynomial time implementation of the KR algorithm on multigraphs, running in the time for O(nm) maximum flow computations, where n (m) is the number of vertices (edges, not counting parallel copies). This complements the implementation of [12] which uses time O((kn)2) and is efficient for small k.