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## Packing Steiner trees

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Citations: | 108 - 5 self |

### Citations

200 |
On the problem of decomposing a graph into n connected factors,
- Tutte
- 1961
(Show Context)
Citation Context ...eorem [8]. The other extreme is when all the vertices are terminals. In this case a Steiner tree is just a spanning tree of the graph, so the problem becomes the classical Nash-Williams-Tutte theorem =-=[9, 11]-=-. Theorem 1 [9, 11] Graph G(V; E) contains k 1 edge-disjoint spanning trees if and only if EG (P)sk(t \Gamma 1) for every partition P = fV 1 ; : : : ; V t g of V into non-empty subsets, where EG (P) d... |

145 | A new greedy approach for facility location problems.
- Jain, Mahdian, et al.
- 2002
(Show Context)
Citation Context ...mma 1 p x 0 + p X i=1 x i + s X i=p+1 i p x i (5) subject to s X i=0 ix i = k s\Gamma1 X i=0 x i = ff s\Gamma1 k 8i : x is0 This is very similar to the idea of using factorrevealing LP's explained in =-=[6, 5]-=-. In order to upper bound the solution of the above linear program, we multiply its first constraint by 1=p 2 and its second constraint by (p \Gamma 1)=p. We obtain the following. s\Gamma1 X i=0 ` p \... |

134 |
Zur allgemeinen Kurventheorie
- Menger
- 1927
(Show Context)
Citation Context ...in combinatorics. One extreme of the problem is when we have only two terminals. In this case a Steiner tree is just a path between the terminals, so the problem becomes the well known Menger theorem =-=[8]-=-. The other extreme is when all the vertices are terminals. In this case a Steiner tree is just a spanning tree of the graph, so the problem becomes the classical Nash-Williams-Tutte theorem [9, 11]. ... |

68 |
The Steiner problem with edge lengths 1
- Bern, Plassmann
- 1989
(Show Context)
Citation Context ... the above reduction preserves the approximation factor of the algorithm. The above theorem together with the algorithm of Hougardy and Promel [10] and APX-completeness proof of Bern and Plassmann [3=-=]-=- for the minimum Steiner tree problem implies the following corollaries. Corollary 4.1. There is a 1.598-approximation algorithm for the fractional Steiner tree packing problem. Corollary 4.2. The fra... |

43 | Packing Steiner trees: a cutting plane algorithm and computational results.
- M, Martin, et al.
- 1996
(Show Context)
Citation Context ...by both practical as well as theoretical considerations. The problem in its full generality (where for each Steiner tree, a dierent set of terminals is given) has applications in VLSI circuit design [=-=15, 9, 8-=-]. In this application, a Steiner tree is needed to share an electric signal by a set of terminal nodes. Another application, which is also our primary focus, arises in the Internet Research supporte... |

38 | A 1.598 Approximation Algorithm for the Steiner Problem in Graphs,”
- Hougardy, Pro¨mel
- 1999
(Show Context)
Citation Context ... R . Furthermore, we know that the optimum solution of the dual program 13, and therefore the primal program 12 is not more than ffR . The above theorem together with the algorithm of Hougardy et al. =-=[4]-=- for the minimum Steiner tree problem implies the following corollary. Corollary 13 There is a 1.598-approximation algorithm for the fractional Steiner tree packing problem. 6 Conclusion In this paper... |

34 | On decomposing a hypergraph into k connected subhypergraphs, Technical report published by the Egrevdry Research Group
- Frank, Kiráby, et al.
- 2001
(Show Context)
Citation Context ...h less is known. In fact, there is no good general non-trivial sufficient bounds, as far as we know. The special case in which V \Gamma S is independent is considered by Kriesell [7] and Frank et al. =-=[3]-=-. A corollary of the main theorem of [7] is that for a graph G(V; E) and S ` V , if V \Gamma S is an independent set and S is k(k + 1)-edge connected, then G contains k edge-disjoint S-Steiner trees. ... |

33 | Randomized metarounding.
- Carr, Vempala
- 2002
(Show Context)
Citation Context ...al program 13 is between R and ffR . The above algorithm computes the approximate value of the solution of the primal program 12. In order to compute the actual solution, we use the technique used in =-=[1]-=-. The total number of separating hyperplanes found by the above separation oracle while running the ellipsoid algorithm for R \Gamma ffl is bounded by a polynomial. These separation oracles are enough... |

31 |
Minimum partition of a matroids into independent sets
- Edmonds
- 1965
(Show Context)
Citation Context ...es of P. Since the problem of finding k disjoint spanning trees in a graph is a special case of finding k disjoint bases of a matroid, Theorem 1 can be derived from Edmonds' matroid partition theorem =-=[2]-=-. Both these theorems, Menger as well as NashWilliam -Tutte theorem, are max-min theorems and can be generalized into a single theorem which says that the maximum number of edge-disjoint Steiner trees... |

24 | Preserving and increasing local edge-connectivity in mixed graphs
- Bang-Jensen, Frank, et al.
- 1995
(Show Context)
Citation Context ...veral parallel edges, one can obtain an ( )-approximation algorithm for the fractional Steiner tree packing problem, for any > 0. It worths mentioning that using Mader's splitting-o lemma (see [2]) one can obtain a combinatorial 2approximation algorithm for the fractional Steiner tree packing problem. The idea is to replace each edge by two parallel edges, and perform the splitting-o procedur... |

23 |
Edge disjoint spanning trees of finite graphs.
- A
- 1961
(Show Context)
Citation Context ...eorem [8]. The other extreme is when all the vertices are terminals. In this case a Steiner tree is just a spanning tree of the graph, so the problem becomes the classical Nash-Williams-Tutte theorem =-=[9, 11]-=-. Theorem 1 [9, 11] Graph G(V; E) contains k 1 edge-disjoint spanning trees if and only if EG (P)sk(t \Gamma 1) for every partition P = fV 1 ; : : : ; V t g of V into non-empty subsets, where EG (P) d... |

8 |
Approximation algorithms for facility location via dual fitting with factor-revealing LP
- Jain, Mahdian, et al.
(Show Context)
Citation Context ...mma 1 p x 0 + p X i=1 x i + s X i=p+1 i p x i (5) subject to s X i=0 ix i = k s\Gamma1 X i=0 x i = ff s\Gamma1 k 8i : x is0 This is very similar to the idea of using factorrevealing LP's explained in =-=[6, 5]-=-. In order to upper bound the solution of the above linear program, we multiply its first constraint by 1=p 2 and its second constraint by (p \Gamma 1)=p. We obtain the following. s\Gamma1 X i=0 ` p \... |

8 |
R.W.: Network information
- Ahlswede, Cai, et al.
- 2000
(Show Context)
Citation Context ... For planar graphs, related problems have been considered by Wagner [20]. A variant of the maximum capacity broadcast problem, in which network encoding is allowed is considered in information theory =-=[1, 13-=-]. In this paper, we will present an algorithm for the Steiner tree packing problem which shows that if a subset S in a graph is k-edge-connected, then there are jSj k edge-disjoint Steiner trees for... |

6 |
Local spanning trees in graphs and hypergraph decomposition with respect to edge connectivity
- Kriesell
- 1999
(Show Context)
Citation Context ...P) denoting the number of edges between distinct classes of P , and jP j denoting the number of classes of P . This statement is not true if we are not in the extreme cases, as shown by an example in =-=[7]-=-. It is an easy exercise to show using Theorem 1 that if a graph G is 2k-edge connected then it has k edge disjoint spanning trees. Therefore, NashWilliam -Tutte's and Menger's theorems give necessary... |

6 | Packing paths and Steiner trees: routing of electronic circuits
- Martin, Weismantel
- 1993
(Show Context)
Citation Context ...by both practical as well as theoretical considerations. The problem in its full generality (where for each Steiner tree, a dierent set of terminals is given) has applications in VLSI circuit design [=-=15, 9, 8-=-]. In this application, a Steiner tree is needed to share an electric signal by a set of terminal nodes. Another application, which is also our primary focus, arises in the Internet Research supporte... |

4 | Bounds on the maximum number of edge-disjoint Steiner trees of a graph, Congressus Numerantium 145
- Petingi, Rodriguez
- 2000
(Show Context)
Citation Context ... in [3], where they weaken the requirement for connectivity of S to 3k-edge connectedness. They also prove some generalizations of Theorem 1 to hypergraphs. For arbitrary set S, Petingi and Rodriguez =-=[10]-=- give a lower bound for the number of edge disjoint S-Steiner trees, by showing that: if S is k-edge connected in G and jSjs2, then G has at least b( 2 3 ) jV \GammaSj kc edge-disjoint S-Steiner trees... |

4 |
The Steiner Tree Packing
- Grotschel, Martin, et al.
- 1997
(Show Context)
Citation Context ...by both practical as well as theoretical considerations. The problem in its full generality (where for each Steiner tree, a dierent set of terminals is given) has applications in VLSI circuit design [=-=15, 9, 8-=-]. In this application, a Steiner tree is needed to share an electric signal by a set of terminal nodes. Another application, which is also our primary focus, arises in the Internet Research supporte... |

2 | Simple algorithms for Steiner trees and paths packing problems in planar graphs
- Wagner
- 1993
(Show Context)
Citation Context ...s, by showing that: if S is k-edge-connected in G and jSj 2, then G has at least b( 2 3 ) jV Sj k=2c edgedisjointsS-Steiner trees. For planar graphs, related problems have been considered by Wagner [=-=20]-=-. A variant of the maximum capacity broadcast problem, in which network encoding is allowed is considered in information theory [1, 13]. In this paper, we will present an algorithm for the Steiner tre... |

1 |
Reconstructing paths. Unpublished manuscript, available online at R. Ravi's homepage
- Conforti, Hassin, et al.
- 1999
(Show Context)
Citation Context ...: we need relatively strong assumptions. Secondly, there is an interesting connection between the algorithm in the proof of Theorem 3.1 and the standard stable marriage algorithm. See Conforti et al. =-=[5]-=- for more details. In the next theorem, we can get a better bound using the idea used in the proof of Theorem 3.1. Theorem 3.2. Let G(V; E) be a graph and S be a subset of s vertices of G. If S is k-e... |

1 |
Beyond rounting: An algebraic approach to network coding
- Koetter, Medard
- 2002
(Show Context)
Citation Context ... For planar graphs, related problems have been considered by Wagner [20]. A variant of the maximum capacity broadcast problem, in which network encoding is allowed is considered in information theory =-=[1, 13-=-]. In this paper, we will present an algorithm for the Steiner tree packing problem which shows that if a subset S in a graph is k-edge-connected, then there are jSj k edge-disjoint Steiner trees for... |

1 | Network information flow - Bang-Jensen, Frank, et al. - 1995 |

1 | Reconstructing flow paths. Unpublished manuscript, available online at R. Ravi's homepage - Conforti, Hassin, et al. - 1999 |