On survivable set connectivity

In the Set Connectivity problem, we are given an n-node edge-weighted undirected graph and a collection of h set pairs (Si, Ti), where Si and Ti are subsets of nodes. The goal is to compute a min-cost subgraph H so that, for each set pair (Si, Ti), there exists at least one path in H between some node in Si and some node in Ti. In this paper, we initiate the study of the Survivable Set Connectivity problem (SSC), i.e., the generalization of Set Connectivity where we are additionally given an integer requirement ki ≥ 1 for each set pair (Si, Ti), and we want to find a min-cost subgraph H so that there are at least ki edge-disjoint paths in H between Si and Ti. We achieve the following main results: • We show that there is no poly-logarithmic approximation for SSC unless NP has a quasi-polynomial time algorithm. This result is based on a reduction from the Minimum Label Cover problem, and the result holds even for the special case where Si = {r} for all i, i.e., for the high-connectivity variant of the classical Group Steiner Tree problem. More precisely, we prove an approximability lower bound of 2log1−ε n for SSC, for any constant ε > 0, which is almost polynomial on n. A technical novelty of our proof is the first use of a padding scheme technique for an edge-connectivity problem on undirected graphs. (Prior to our results, the applications of this technique only pertain to either node-connectivity problems or problems on directed graphs). • We present a bicriteria approximation algorithm for SSC that computes a solution H of cost at most poly-logarithmically larger than the optimal cost and provides a connectivity at least Ω(ki/log n) for each set pair (Si, Ti). The main algorithmic idea is to solve a standard LP relaxation to the problem, and then embed the resulting fractional capacities into a tree via Räcke's cut-based tree embeddings. Based on that, we generate a random collection of Group Steiner Tree-like fractional solutions, which can then be handled by the rounding scheme of [Garg, Konjevod and Ravi -- SODA'98]. The prior work on Set Connectivity and Group Steiner Tree used Bartal's distance-based tree embeddings which do not seem to generalize to the k-connectivity versions of these problems. Finally, we remark an interesting contrast demonstrated by our results: While our hardness result almost rules out \"polynomial\" approximation ratios, relaxing connectivity constraints allows us to obtain \"poly-logarithmic\" bounds. This naturally suggests that relaxing connectivity requirements might be a proper way in getting big improvements, even beyond (non-bicriteria) lower bounds, for other connectivity problems, especially those whose approximability lower bounds are derived from Minimum Label Cover.