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 D = {(S ₁,T ₁),... ,(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 ⊆ G connects 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, 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 O(log ² n log ² 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(log ³ n log 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+∈) 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 polylogarithmic approximation; this result improves on the previously known ratio which can be Ω(n) in the worst case.