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

In the generalized connectivity problem, we are given an edge-weighted graph G = (V, E) and a collection = {(S₁, T₁), ⋯ , (S_k, T_k)} of distinct demands; each demand (S_i , T_i ) is a pair of disjoint vertex subsets. We say that a subgraph F of Gconnects a demand (S_i , T_i ) when it contains a path with one endpoint in S_i and the other in T_i . The goal is to identify a minimum weight subgraph that connects all demands in D. 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, facility location with nonmetric costs, tree multicast, and group Steiner tree. Obtaining a nontrivial approximation ratio for generalized connectivity was left as an open problem. We describe the first poly-logarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O(log² nlog² k). Here, n is the number of vertices in G and k is the number of demands. We also prove that the cut-covering relaxation of this problem has an O(log³ nlog ² k) 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 directed Steiner network problem, we obtain an O(k^1/2+ε ) approximation which improves on the currently best performance guarantee of Õ(k^2/3) due to Charikar et al. (SODA '98). For the set connector problem, recently introduced by Fukunaga and Nagamochi (IPCO '07), we present a poly-logarithmic approximation; this result improves on the previously known ratio which can be Ω(n) in the worst case.