## Packing Steiner trees

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Citations: | 88 - 5 self |

### BibTeX

@MISC{Jain_packingsteiner,

author = {Kamal Jain and Mohammad R. Salavatipoury and et al.},

title = {Packing Steiner trees},

year = {}

}

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### Abstract

The Steiner packing problem is to find the maximum number of edge-disjoint subgraphs of a given graph G that connect a given set of required points S. This problem is motivated by practical applications in VLSI-layout and broadcasting, as well as theoretical reasons. In this paper, we study this problem and present an algorithm with an asymptotic approximation factor of |S|/4. This gives a sufficient condition for the existence of k edge-disjoint Steiner trees in a graph in terms of the edge-connectivity of the graph. We will show that this condition is the best possible if the number of terminals is 3. At the end, we consider the fractional version of this problem, and observe that it can be reduced to the minimum Steiner tree problem via the ellipsoid algorithm.

### Citations

133 |
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... |

120 | 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 \... |

83 |
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]. ... |

57 |
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... |

32 | A 1.598 approximation algorithm for the steiner problem in graphs
- Hougardy, Prömel
- 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... |

31 | Packing Steiner trees: A cutting plane algorithm and computational results
- Grötschel, 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... |

25 |
Minimum partition of a matroid 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 | Randomized meta-rounding
- Carr, Vempala
(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... |

23 | On decomposing a hypergraph into k connected subhypergraphs
- Frank, Király, et al.
(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. ... |

23 | Preserving and increasing local edgeconnectivity 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... |

18 |
Nash-Williams, 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... |

9 |
Approximation algorithms for facility location via dual fitting with factor-revealing LP
- 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 \... |

5 |
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... |

5 | 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 |
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... |

3 | Bounds on the maximum number of edge-disjoint Steiner trees of a graph
- 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... |

3 |
The Steiner Tree Packing
- Grötschel, 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 |