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 1.85. We prove the KR algorithm has approximation ratio ≤ 1 + √1/e 1.61; for odd 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.
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机译：Khuller和Raghavachari [12]提出了一种近似算法（ KR算法），用于找到最小的 k 边连接的跨子图（ k-ECSS ）的无向多重图。他们证明了KR算法的近似比<1.85。我们证明了KR算法的近似比≤1 +√1/ e <1.61;对于奇数 k ，这需要对算法进行较小的修改。这是针对任意 k 的最小 k -ECSS问题的最著名的性能界限。我们的分析还给出了对于 k ≤3的任何固定值的最著名的性能范围，例如，对于偶数 k 的近似比率为≤1 +（1-1 / k ） k / 2 。我们的分析基于层状集合集（类似于相关上下文中使用的集合），它可以更好地说明算法的先前迭代中添加的边。我们还提出了KR算法在多图上的多项式时间实现，该时间用于 O （ nm ）最大流量计算的时间，其中<I> n （ m ）是顶点数（边，不计算平行副本）。这是对[12]的实现的补充，该实现使用时间 O （（（ kn ） 2 ）），并且对于较小的 k I>。
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