## Abstract

In the generalized connectivity problem, we are given an edge-weighted graph G = (V, E) and a collection = {(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 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(log^{2} nlog^{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} 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.

Original language | English |
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Article number | 18 |

Journal | ACM Transactions on Algorithms |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2011 |

## Keywords

- Approximation algorithms
- Directed Steiner network
- Generalized connectivity

## ASJC Scopus subject areas

- Mathematics (miscellaneous)