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## A polylogarithmic approximation algorithm for the group Steiner tree problem (2000)

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### Other Repositories/Bibliography

Venue: | Journal of Algorithms |

Citations: | 146 - 9 self |

### Citations

1958 | the probabilistic method.
- Alon, Spencer
- 2008
(Show Context)
Citation Context ...say e,e,... ,p are picked in their respective independent random experiments. The probability of this is x e__. x e__ ... __ = x. [] To analyze this experiment, we use Janson's inequality (see, e.g., =-=[1]-=-), which can be stated as follows: let fl be a universal set, and R C_ fl determined by the experiment in which each element r E fl is independently included in R with probability Pt. Let Ai be subset... |

1394 | Geometric algorithms and combinatorial optimization,” - Grotschel, Lovasz, et al. - 1987 |

766 | A threshold of ln n for approximating set cover,
- Feige
- 1998
(Show Context)
Citation Context ...g a set by attaching as many children to it as its size, one for each element, using zero-weight edges). Because of this, it is NP-hard to approximate the group Steiner problem to a factor of o(ln k) =-=[8, 20, 3]-=-. 1.2 Previous Work. The papers of Ihler [10, 11, 12], and Ihler, Reich and Widmayer [13, 14] contain some early work on the group Steiner problem. (In some of the cited papers the group Steiner probl... |

390 | Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
- Arora
- 1998
(Show Context)
Citation Context ...tifying some points we can assume the distances to be in (1,... , O(n2)). This can be done so that the optimum value of a group Steiner tree only changes by a factor of 1 + e for any constant e as in =-=[2]-=-. 6.2 Service constrained network design problems. Marathe et al. [17, 18] study the following problem: given an undirected graph G = (V, E) with two different cost functions c (modeling the service c... |

369 |
Randomized rounding: A technique for provably good algorithms and algorithmic proofs.
- Raghavan, Thompson
- 1987
(Show Context)
Citation Context ...ective function. $ Random experiment. In this section we explain our rounding process and prove the main technical results. Our rounding may be seen as an extension of traditional randomized rounding =-=[19] for the s-=-et cover problem to our "tree version" of the set-covering problem. Consider the following random experiment. Assume without loss of generality that all group vertices are leaves of T' (inte... |

366 | Approximation algorithms for connected dominating sets,” - Guha, Khuller - 1998 |

350 | Probabilistic approximations of metric spaces and its algorithmic applications
- Bartal
- 1996
(Show Context)
Citation Context ... problem on trees (even this is still as hard as set-covering, by the reduction described above) with a O(log k log N) approximation ratio. This is extended to arbitrary graphs by using the result of =-=[4]-=-, and the final approximation ratio is then O(log 3 n log k) (the size of the largest group, N, is at most the number of nodes n). The results of [16] used in place of Bartal's improve the performance... |

322 | A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP.
- Raz, Safra
- 1997
(Show Context)
Citation Context ...g a set by attaching as many children to it as its size, one for each element, using zero-weight edges). Because of this, it is NP-hard to approximate the group Steiner problem to a factor of o(ln k) =-=[8, 20, 3]-=-. 1.2 Previous Work. The papers of Ihler [10, 11, 12], and Ihler, Reich and Widmayer [13, 14] contain some early work on the group Steiner problem. (In some of the cited papers the group Steiner probl... |

265 | On approximating arbitrary metrics by tree metrics,” - Bartal - 1998 |

172 | Steiner problem in networks: a survey. - Winter - 1987 |

172 | Approximation algorithms for directed steiner problems - Charikar, Chekuri, et al. - 1999 |

144 |
There are planar graphs almost as good as the complete graph
- Chew
- 1989
(Show Context)
Citation Context ...s that exclude Ks,s as a minor for some fixed constant s. An example is planar graphs that exclude Ka,s. Since planar graph distances approximate distances in the two-dimensional Euclidean plane well =-=[7]-=-, the improvement also carries over to group Steiner problems in the plane. Our approximation algorithm for the case of tree metrics first solves a linear programming relaxation of the group Steiner t... |

141 | Improved low degree testing and its applications.
- Arora, Sudan
- 2003
(Show Context)
Citation Context ...g a set by attaching as many children to it as its size, one for each element, using zero-weight edges). Because of this, it is NP-hard to approximate the group Steiner problem to a factor of o(ln k) =-=[8, 20, 3]-=-. 1.2 Previous Work. The papers of Ihler [10, 11, 12], and Ihler, Reich and Widmayer [13, 14] contain some early work on the group Steiner problem. (In some of the cited papers the group Steiner probl... |

133 | A nearly best-possible approximation algorithm for node-weighted steiner trees
- Klein, Ravi
- 1995
(Show Context)
Citation Context ... generalization of the classical Steiner tree problem [23], and therefore NPhard. In fact, it is also a direct generalization of the even harder set-covering problem as observed several times earlier =-=[11, 15, 22]-=-. In the set covering problem, we are given a collection of weighted subsets of a given ground set and seek a minimum-weight subcollection whose union is the entire ground set. To reduce this problem ... |

86 | Bicriteria network design problems,”
- Marathe, Ravi, et al.
- 1998
(Show Context)
Citation Context ...ility, we use Janson's inequality. Our algorithm works with similar performance bounds when applied to the errand scheduling problem of [22], and to the service-constrained network design problems of =-=[17, 18]-=-. In the remainder of the paper, we first present our linear programming formulation for the problem on tree metrics and our rounding procedure, and then prove the performance guarantee. Then, we desc... |

76 | A Branch-and-Cut Algorithm for the Symmetric Generalized Travelling Salesman Problem", working paper - Fischetti, Salazar, et al. - 1993 |

73 |
Beyond Steiner’s problem: a VLSI oriented generalization
- Reich, Widmayer
- 1990
(Show Context)
Citation Context ...id for the group Steiner problem on planasgraphs as well as on a set of points in the 2D-Euclidean case. 1 Introduction. 1.1 Motivation. The group Steiner problem was introduced by Reich and Widmayer =-=[21]-=-. The problem arises in wire routing with multiport terminals in physical VLSI design. The traditional model assuming single ports for each of the terminals to be connected in a net of minimum length ... |

60 |
Steiner problem in networks { a survey. Networks 17
- Winter
- 1987
(Show Context)
Citation Context ...n to the neighbors in the clique, every copy of this node has the same neighbors as the original node in the graph. The group Steiner problem is a generalization of the classical Steiner tree problem =-=[23]-=-, and therefore NPhard. In fact, it is also a direct generalization of the even harder set-covering problem as observed several times earlier [11, 15, 22]. In the set covering problem, we are given a ... |

50 | Poisson approximation for large deviations - Janson - 1990 |

26 | Bounds on the quality of approximate solutions to the group steiner problem
- IHLER
- 1991
(Show Context)
Citation Context ...size, one for each element, using zero-weight edges). Because of this, it is NP-hard to approximate the group Steiner problem to a factor of o(ln k) [8, 20, 3]. 1.2 Previous Work. The papers of Ihler =-=[10, 11, 12]-=-, and Ihler, Reich and Widmayer [13, 14] contain some early work on the group Steiner problem. (In some of the cited papers the group Steiner problem is referred to as the class Steiner problem, or th... |

24 | On approximating planar metrics by tree metrics
- Konjevod, Ravi, et al.
- 2001
(Show Context)
Citation Context ...xtended to arbitrary graphs by using the result of [4], and the final approximation ratio is then O(log 3 n log k) (the size of the largest group, N, is at most the number of nodes n). The results of =-=[16]-=- used in place of Bartal's improve the performance ratio to O(log 2 n log k) on graphs that exclude Ks,s as a minor for some fixed constant s. An example is planar graphs that exclude Ka,s. Since plan... |

23 | Provably good routing tree construction with multiport terminals
- BATEMAN, HELVIG, et al.
- 1997
(Show Context)
Citation Context ...io of BsH(N) = BsO(lnN), where B is the maximum number of vertices of a group in a subtree of the root, and H(N) is the N-th harmonic number. In a recent paper, Bateman, Helvig, Robins and Zelikovsky =-=[6]-=- have given the first algorithm with a sublinear performance guarantee. Their algorithm (with a Java implementation available on the Internet [5]) gives an approximation ratio of (1 + 1_)v/. This rati... |

18 | The record balancing problem: A dynamic programming solution of the generalized traveling salesman problem - Henry-Labordere - 1969 |

9 |
Geometric Algorithms and Combinatorial Optimization
- GrStschel, Lovdsz, et al.
- 1988
(Show Context)
Citation Context ... Sgi = 0 for some i 0_x_ 1, VeEE where rsV is the root vertex. The above linear program can be solved in polynomial time, despite the exponential number of constraints. This follows, for example from =-=[9]-=- and the fact that a separation oracle can be constructed easily, using a minimum cut procedure. A more direct way to see the polynomial-time solvability of the program is by re-interpreting the const... |

9 | The complexity of approximating the class Steiner tree problem
- Ihler
- 1991
(Show Context)
Citation Context ... generalization of the classical Steiner tree problem [23], and therefore NPhard. In fact, it is also a direct generalization of the even harder set-covering problem as observed several times earlier =-=[11, 15, 22]-=-. In the set covering problem, we are given a collection of weighted subsets of a given ground set and seek a minimum-weight subcollection whose union is the entire ground set. To reduce this problem ... |

8 | On shortest networks for classes of points in the plane
- Ihler, Reich, et al.
- 1991
(Show Context)
Citation Context ...t edges). Because of this, it is NP-hard to approximate the group Steiner problem to a factor of o(ln k) [8, 20, 3]. 1.2 Previous Work. The papers of Ihler [10, 11, 12], and Ihler, Reich and Widmayer =-=[13, 14]-=- contain some early work on the group Steiner problem. (In some of the cited papers the group Steiner problem is referred to as the class Steiner problem, or the tree-cover problem.) In particular, in... |

6 | Class steiner trees and vlsi-design - Ihler, Reich, et al. - 1999 |

6 | Mathematical model of scheduling clients through welfare agencies - Saksena - 1970 |

5 | The rectilinear class Steiner tree problem for intervals on two parallel lines
- Ihler
- 1994
(Show Context)
Citation Context ...size, one for each element, using zero-weight edges). Because of this, it is NP-hard to approximate the group Steiner problem to a factor of o(ln k) [8, 20, 3]. 1.2 Previous Work. The papers of Ihler =-=[10, 11, 12]-=-, and Ihler, Reich and Widmayer [13, 14] contain some early work on the group Steiner problem. (In some of the cited papers the group Steiner problem is referred to as the class Steiner problem, or th... |

4 | Service-constrained network design problems - Marathe, Ravi, et al. - 1996 |

2 |
Improved results on service-constrained network design problems. to appear
- Marathe, Ravi, et al.
- 1997
(Show Context)
Citation Context ...ility, we use Janson's inequality. Our algorithm works with similar performance bounds when applied to the errand scheduling problem of [22], and to the service-constrained network design problems of =-=[17, 18]-=-. In the remainder of the paper, we first present our linear programming formulation for the problem on tree metrics and our rounding procedure, and then prove the performance guarantee. Then, we desc... |

2 | On the traveling purchaser problem - Ravi, Salman - 1999 |

2 | The Errand Scheduling Problem - Slavk - 1997 |

1 |
Java program, available at http ://www. cs. virginia. edu/robins/groupSt einer
- Bateman, Helvig, et al.
- 1997
(Show Context)
Citation Context ...a recent paper, Bateman, Helvig, Robins and Zelikovsky [6] have given the first algorithm with a sublinear performance guarantee. Their algorithm (with a Java implementation available on the Internet =-=[5]-=-) gives an approximation ratio of (1 + 1_)v/. This ratio comes from approximating the group Steiner tree by a 2-star (tree of depth 2), and then approximating the 2-star within a logarithmic factor. 1... |

1 |
The errand scheduling problem. manuscript, available at http://win,. cs
- Slav
- 1996
(Show Context)
Citation Context ... generalization of the classical Steiner tree problem [23], and therefore NPhard. In fact, it is also a direct generalization of the even harder set-covering problem as observed several times earlier =-=[11, 15, 22]-=-. In the set covering problem, we are given a collection of weighted subsets of a given ground set and seek a minimum-weight subcollection whose union is the entire ground set. To reduce this problem ... |

1 | The errand scheduling problem. manuscript, available at http://www.cs.buffalo.edu/ slavik/papers - Slav'ik - 1996 |