On Survivable Set Connectivity
Parinya Chalermsook, Fabrizio Gr, Oni Bundit Laekhanukit · 2015
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 approx-imation for SSC unless NP has a quasi-polynomial time algorithm. This result is based on a reduc-tion 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 prob-lem. More precisely, we prove an approximability lower bound of 2log 1− n for SSC, for any constant > 0, which is almost polynomial on n. A techni-cal 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 ei-ther node-connectivity problems or problems on di-rected graphs). • We present a bicriteria approximation algorithm for SSC that computes a solution H of cost at most ∗This work was partially done while the first and third authors