Set connectivity problems in undirected graphs and the directed Steiner network problem

In the <i>generalized connectivity</i> problem, we are given an edge-weighted graph <i>G = (V, E)</i> and a collection <i>D</i> = {(S<sub>1</sub>,T<sub>1</sub>),…, (S<sub>k</sub>,T<sub>k</sub>)} of distinct <i>demands</i>; each demand (S<sub>i</sub>, T<sub>i</sub>) is a pair of disjoint vertex subsets. We say that a subgraph <i>F ⊆ G connects</i> a demand (S<sub>i</sub>, T<sub>i</sub>) when it contains a path with one endpoint in <i>S<sub>i</sub></i> and the other in T<sub>i</sub>. The goal is to identify a minimum weight subgraph that connects all demands in <i>D</i>. Alon et al. (SODA '04) introduced this problem to study online network formation settings and showed that it captures some well-studied problems such as Steiner forest, non-metric facility location, tree multicast, and group Steiner tree. Finding a non-trivial approximation ratio for generalized connectivity was left as an open problem. Our starting point is the first polylogarithmic approximation for generalized connectivity attaining a performance guarantee of <i>O(log<sup>2</sup> n log<sup>2</sup> k)</i>. Here <i>n</i> is the number of vertices in G and <i>k</i> is the number of demands. We also prove that the cut-covering relaxation of this problem has an <i>O(log<sup>3</sup> n log<sup>2</sup> k)</i> integrality gap. Building upon the results for generalized connectivity we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the <i>directed Steiner network</i> problem, we obtain an <i>O</i>(k<sup>1/2+ε</sup>) approximation, which improves on the currently best performance guarantee of O(k<sup>2/3</sup>) due to Charikar et al. (SODA '98). For the <i>set</i> <i>connector</i> problem, recently introduced by Fukunaga and Nagamochi (IPCO '07), we present a polylogarithmic approximation; this result improves on the previously known ratio which can be Ω(<i>n</i>) in the worst case.

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