Approximation Algorithms for (S,T)-Connectivity Problems

We study a directed network design problem called the k-(S, T )-connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer k, the minimum cost k-vertex connected spanning subgraph problem is a special case of the k(S, T )-connectivity problem. We defer precise statements of the problem and of our results to the introduction. For k = 1, we call the problem the (S, T )-connectivity problem. We study three variants of the problem: the standard (S, T )-connectivity problem, the relaxed (S, T )-connectivity problem, and the unrestricted (S, T )-connectivity problem. We give hardness results for these three variants. We design a 2-approximation algorithm for the standard (S, T )-connectivity problem. We design tight approximation algorithms for the relaxed (S, T )-connectivity problem and one of its special cases. For any k, we give an O(log k log n)-approximation algorithm, where n denotes the number of vertices. The approximation guarantee almost matches the best approximation guarantee known for the minimum cost k-vertex connected spanning subgraph problem which is O(log k log n n−k ) due to Nutov in 2009 [62].

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