Abstract
Consider the minimum sizek-edge-connected spanning subgraph problem: given a positive integerkand ak-edge-connected graphG, find ak-edge-connected spanning subgraph ofGwith the minimum number of edges. This problem is known to be NP-complete. Khuller and Raghavachari presented the first algorithm which, for allk, achieves a performance ratio smaller than a constant which is less than two. They proved an upper bound of 1.85 for the performance ratio of their algorithm. Currently, the best known performance ratio for the problem is 1+2/(k+1), achieved by a slower algorithm of Cheriyan and Thurimella. In this article, we improve Khuller and Raghavachari's analysis, proving that the performance ratio of their algorithm is smaller than 1.7 for large enoughk, and that it is at most 1.75 for allk. Second, we show that the minimum size 2-edge-connected spanning subgraph problem is MAX SNP-hard.
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