## An Improved Approximation Ratio for the Covering Steiner Problem. On the Covering Steiner problem (2006)

Venue: | Theory of Computing |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Gupta06animproved,

author = {Anupam Gupta and Aravind Srinivasan},

title = {An Improved Approximation Ratio for the Covering Steiner Problem. On the Covering Steiner problem},

journal = {Theory of Computing},

year = {2006},

volume = {2},

pages = {2006}

}

### OpenURL

### Abstract

Abstract: In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems; indeed, when all the vertices of the graph lie in one group with a requirement of k, we get the k-MST problem, and when there are multiple groups with unit requirements, we obtain the Group Steiner problem. While many covering problems (e.g., the covering integer programs such as set cover) become easier to approximate as the requirements increase, the Covering Steiner problem

### Citations

129 | A polylogarithmic approximation algorithm for the group Steiner tree problem.Journal of Algorithms
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(Show Context)
Citation Context ...ing Steiner problem given by Even et al. [3] and Konjevod et al. [11]. Our results match the approximation guarantee of the known randomized algorithm for the Group Steiner problem due to Garg et al. =-=[6]-=- (see the paper of Charikar et al. [2] for a deterministic algorithm). A suitable melding of a randomized rounding approach with a deterministic “threshold rounding” method leads to our result. Let G ... |

53 |
Saving an epsilon: a 2-approximation for the k-MST problem in graphs
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Citation Context ...ne can get an O(k) approximation algorithm using well-known ideas without resorting to the reduction to tree instances outlined above. Indeed, one can use the 2-approximation algorithm (given by Garg =-=[5]-=-) for the rooted ri-MST on each group gi separately, and return the union of the k trees obtained. However, in case the number of groups k is large, one may prefer an algorithm whose dependence on the... |

50 |
Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics
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Citation Context ...a tree, since the notion of probabilistic tree embeddings [1] can be used to reduce an arbitrary instance of the problem to a instance on a tree. Specifically, via the result of Fakcharoenphol et al. =-=[4]-=-, a ρ– approximation algorithm on tree-instances implies an O(ρ logn)–approximation algorithm for arbitrary instances. In fact, we can assume that the instance is a rooted tree instance where the root... |

36 |
Poisson approximation for large deviations
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Citation Context ...ed rounding scheme is expected to fully cover a constant fraction of the groups. A chip game presented in Section 3 is used to bound the number of iterations where case (i) holds; Janson’s inequality =-=[9]-=- and some deterministic arguments are used for case (ii). In the interest of the cleanest exposition, we have not attempted to optimize our constants. 2 Preliminaries The analysis of our algorithm wil... |

28 | Integrality ratio for group steiner trees and directed steiner trees
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Citation Context ...ld like to note that the integrality gap of this relaxation for tree-instances of Group Steiner is Ω(logk · THEORY OF COMPUTING, Volume 2 (2006), pp. 53–64 55sA. GUPTA AND A. SRINIVASAN logN/loglogN) =-=[7]-=-. Since this integrality gap naturally extends to the Covering Steiner problem as well, this suggests that our techniques cannot improve the approximation guarantee substantially. 1.1 Our Techniques O... |

24 | New approaches to covering and packing problems - SRINIVASAN - 2001 |

23 | On network design problems: fixed cost flows and the Covering Steiner problem
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- 2002
(Show Context)
Citation Context ...as for the case of all unit requirements, which is just the Group Steiner problem. In this work, we improve on the known approximation algorithms for the Covering Steiner problem given by Even et al. =-=[3]-=- and Konjevod et al. [11]. Our results match the approximation guarantee of the known randomized algorithm for the Group Steiner problem due to Garg et al. [6] (see the paper of Charikar et al. [2] fo... |

11 |
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Citation Context ...l. [3] and Konjevod et al. [11]. Our results match the approximation guarantee of the known randomized algorithm for the Group Steiner problem due to Garg et al. [6] (see the paper of Charikar et al. =-=[2]-=- for a deterministic algorithm). A suitable melding of a randomized rounding approach with a deterministic “threshold rounding” method leads to our result. Let G = (V,E) be an undirected graph with a ... |

1 |
BARTAL: Probabilistic approximations of metric spaces and its algorithmic applications
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(Show Context)
Citation Context ...ll be to the base two unless specified otherwise. As in the paper of Garg et al. [6], we focus on the case where the given graph G = (V,E) is a tree, since the notion of probabilistic tree embeddings =-=[1]-=- can be used to reduce an arbitrary instance of the problem to a instance on a tree. Specifically, via the result of Fakcharoenphol et al. [4], a ρ– approximation algorithm on tree-instances implies a... |

1 |
AND ROBERT KRAUTHGAMER: Polylogarithmic inapproximability
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(Show Context)
Citation Context ...s small, we would have a good approximation algorithm anyway.) The bounds for tree instances are essentially the best possible in the following asymptotic sense: the paper of Halperin and Krauthgamer =-=[8]-=- shows that, for any constant ε > 0, an O((log(n + k)) 2−ε )approximation algorithm for the Covering Steiner problem implies that NP ⊂ ZTIME[exp{(logn) O(1) }]. (Technically, the hardness is shown for... |

1 |
KHANDEKAR: Lagrangian Relaxation based Algorithms for Convex Programming Problems
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Citation Context ...ntly lose a logarithmic factor due to the step of approximating general graphs by distributions over tree metrics. As a practical matter, it would be interesting to employ frameworks such as those of =-=[10]-=- to approximately solve the linear program by combinatorial means, or to develop a new combinatorial approximation algorithm matching our bounds. Acknowledgments We thank Eran Halperin, Goran Konjevod... |

1 |
AND ARAVIND SRINIVASAN: Approximation algorithms for the covering Steiner problem. Random Structures and Algorithms, 20(3):465–482, 2002. Special Issue on Probabilistic methods in combinatorial optimization
- KONJEVOD, RAVI
(Show Context)
Citation Context ...nit requirements, which is just the Group Steiner problem. In this work, we improve on the known approximation algorithms for the Covering Steiner problem given by Even et al. [3] and Konjevod et al. =-=[11]-=-. Our results match the approximation guarantee of the known randomized algorithm for the Group Steiner problem due to Garg et al. [6] (see the paper of Charikar et al. [2] for a deterministic algorit... |