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## Placing regenerators in optical networks to satisfy multiple sets of requests (2012)

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Venue: | IEEE Transactions on Networking |

Citations: | 3 - 2 self |

### Citations

14063 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ... 1); they can be found in [16].2 Preliminaries Placing Regenerators to Satisfy Multiple Sets of Requests 5 We use standard terminology concerning graphs, complexity, and algorithms; see for instance =-=[7, 11, 21]-=-, respectively. Graphs. All the graphs considered in this article are simple and undirected. Given a graph G we denote by V (G) and E(G) the sets of vertices and edges of G, respectively. If H is a su... |

1278 |
Approximation Algorithms
- Vazirani
- 2003
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Citation Context ... 1); they can be found in [16].2 Preliminaries Placing Regenerators to Satisfy Multiple Sets of Requests 5 We use standard terminology concerning graphs, complexity, and algorithms; see for instance =-=[7, 11, 21]-=-, respectively. Graphs. All the graphs considered in this article are simple and undirected. Given a graph G we denote by V (G) and E(G) the sets of vertices and edges of G, respectively. If H is a su... |

1211 | Parameterized Complexity
- Downey, Fellows
- 1999
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Citation Context ...o better understand its complexity. A natural choice for a parameter could be the number of regenerators used by a feasible solution. The powerful techniques of the theory of parameterized complexity =-=[32]-=- could be very helpful in designing efficient and practical algorithms for finding optimal solutions in real networks, even if -TR is NP-hard. Finally, we assumed that we are given a discrete (finite ... |

1198 | Graph Theory
- Diestel
- 1997
(Show Context)
Citation Context ... 1); they can be found in [16].2 Preliminaries Placing Regenerators to Satisfy Multiple Sets of Requests 5 We use standard terminology concerning graphs, complexity, and algorithms; see for instance =-=[7, 11, 21]-=-, respectively. Graphs. All the graphs considered in this article are simple and undirected. Given a graph G we denote by V (G) and E(G) the sets of vertices and edges of G, respectively. If H is a su... |

375 |
Approximation algorithms for NP-complete problems on planar graphs.
- BAKER
- 1994
(Show Context)
Citation Context ... results of [10] for d = 2, where it is proved that for any d ≥ 2, d-RL is polynomial-time solvable in trees and rings. Finally, it also follows from Remark 1 that 2-RL admits a PTAS in planar graphs =-=[4]-=- and, more generally, in any family of minor-free graphs [6]. 6.2 Bounded number of regenerators per vertex From a technological point of view, it makes sense to introduce a parameter k that limits th... |

84 |
Some APX-completeness results for cubic graphs,
- Alimonti, Kann
- 2000
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Citation Context ...has maximum degree at most 3 and the lightpaths have length O(d). Before this, we need two technical results to be used in the reductions. Minimum Vertex Cover is known to be APX-hard in cubic graphs =-=[1]-=-. By a simple reduction, we prove in the following lemma that Minimum Vertex Cover is also APX-hard in a class of graphs with degree at most 3 and high girth, which will be used in the sequel.6 G.B. ... |

56 | Algorithmic graph minor theory: Decomposition, approximation, and coloring.
- Demaine, Hajiaghayi, et al.
- 2005
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Citation Context ...d ≥ 2, d-RL is polynomial-time solvable in trees and rings. Finally, it also follows from Remark 1 that 2-RL admits a PTAS in planar graphs [4] and, more generally, in any family of minor-free graphs =-=[6]-=-. 6.2 Bounded number of regenerators per vertex From a technological point of view, it makes sense to introduce a parameter k that limits the number of regenerators that can be used at a single vertex... |

41 | Approximation of k-Set Cover by Semi-Local Optimization.
- Duh, Furer
- 1997
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Citation Context ...f the Minimum Set Cover instance is made of a vertex v ∈ V (G) together with a choice of at most one lightpath from each Li that goes through v. The number of possible sets for each v is at most ∏p i=1 |Li|, so the total number of sets is at most |V (G) |· ∏p i=1 |Li|, which is polynomial in the input size since p is fixed. Each such set has cost 1, which corresponds to the cost of a regenerator placed at a vertex and used by these lightpaths. Clearly, a set cover of minimum cost corresponds to a placement of the minimum number of regenerators d-satisfying all the lightpaths. The algorithm of [10] achieves an approximation ratio of Hk − 1/2, where k is the maximum size of a set and Hk = ∑k i=1 1 i . In our case, each regenerator covers at most d edges of at most p lightpaths, so the size of the sets is at most d · p. Therefore, the approximation ratio of the algorithm is at most Hd·p − 1/2. Note that for big d, p, Hd·p ≈ ln d + ln p + 1/2, so comparing both approximation ratios, we have that p < ln d+ ln p when d = Ω(2p). 5 The case of the path In this section we focus on the case where the network topology is a path, which is one of the most important topologies in real networks, as... |

30 | Sparse Regeneration in Translucent WavelengthRouted Optical Networks: Architecture, Network Design and Wavelength Routing,” Photonic Network Communications,
- Yang
- 2005
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Citation Context ...evices in the case when the network is a path and the fiber carries 3 wavelengths. The problem of placing regenerators in optical networks has attracted the attention of several recent research works =-=[5, 8, 9, 13, 18, 19, 22, 23]-=-. Mostly, these articles propose heuristics and run simulations in order to reduce the number of regenerators, but no theoretical analysis is presented. Recently, the first theoretical study of the pr... |

23 | Dynamic routing in translucent WDM optical networks.
- Yang, Ramamurthy
- 2002
(Show Context)
Citation Context ...evices in the case when the network is a path and the fiber carries 3 wavelengths. The problem of placing regenerators in optical networks has attracted the attention of several recent research works =-=[5, 8, 9, 13, 18, 19, 22, 23]-=-. Mostly, these articles propose heuristics and run simulations in order to reduce the number of regenerators, but no theoretical analysis is presented. Recently, the first theoretical study of the pr... |

21 | Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs.
- Austrin, Khot, et al.
- 2009
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Citation Context ..., by reduction from Minimum Vertex Cover in cubic graphs. It would be interesting to determine which is the explicit approximation lower bound given by Theorem 1. The recent results of Austrin et al. =-=[3]-=- about the hardness of Minimum Vertex Cover in graphs of bounded degree may shed some light on this question. We provided an approximation algorithm for (d, p)-TR with constant ratio ln(d · p), by red... |

21 |
Two-Coloring the Edges of a Cubic Graph Such That Each Monochromatic Component Is a Path of Length at Most 5.
- Thomassen
- 1999
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Citation Context ...which will be used in the sequel.6 G.B. Mertzios et al. Lemma 1. Minimum Vertex Cover is APX-hard in the class of graphs H obtained from cubic graphs by subdividing each edge twice. Thomassen proved =-=[20]-=- that the edges of any cubic graph can be two-colored such that each monochromatic connected component is a path of length at most 5. In addition, the aforementioned coloring can be found in polynomia... |

16 | The regenerator location problem.
- Chen, Raghavan
- 2007
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Citation Context ...evices in the case when the network is a path and the fiber carries 3 wavelengths. The problem of placing regenerators in optical networks has attracted the attention of several recent research works =-=[5, 8, 9, 13, 18, 19, 22, 23]-=-. Mostly, these articles propose heuristics and run simulations in order to reduce the number of regenerators, but no theoretical analysis is presented. Recently, the first theoretical study of the pr... |

14 |
On the complexity of the regenerator placement problem in optical networks.
- Flammini, Marchetti-Spaccamela, et al.
- 2009
(Show Context)
Citation Context ...e are no more than d hops without meeting a regenerator. While most of the research has focused on heuristics and simulations, the first theoretical study of the problem has been recently provided in =-=[10]-=-, where the considered cost function is the number of locations in the network hosting regenerators. Nevertheless, in many situations a more accurate estimation of the real cost of the network is give... |

13 | Hardness and Approximation of Traffic Grooming.
- Amini, Perennes, et al.
- 2009
(Show Context)
Citation Context ...exity results when the routing of the requests is not given). We also assume that lightpaths sharing an edge use different wavelengths. That is, we deal with optical networks without traffic grooming =-=[2]-=-. The length of a lightpath is the number of edges it contains. We consider symmetric lightpaths, that is, a lightpath with endpoints u and v consists of a request from u to v and a request from v to ... |

11 |
Regenerator placement algorithms for connection establishment in all-optical networks.
- Kim, Seo
- 2001
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11 |
On linear k-arboricity.
- Bermond, Fouquet, et al.
- 1984
(Show Context)
Citation Context ... (H) such that |SH |≤ (1 + ε) ·OPTVC(H). By the above discussion, we can find a solution SG ⊆ V (G) such that |SG |≤ |SH |− |E(G) |≤ (1 + ε) ·OPTVC(H)− |E(G)| = (1 + ε) · (OPTVC(G) + |E(G)|)− |E(G)| = (1 + ε) ·OPTVC(G) + ε · |E(G) |≤ (1 + ε) ·OPTVC(G) + 3ε ·OPTVC(G) = (1 + 4ε) ·OPTVC(G) , where we have used Equation (1) and the fact that |E(G) |≤ 3 · OPTVC(G). That is, the existence of a PTAS for Minimum Vertex Cover in the class of graphs H would imply the existence of a PTAS in the class of cubic graphs, which is a contradiction by [1] unless P = NP. Solving a conjecture of Bermond et al. [5], Thomassen proved [22] that the edges of any cubic graph can be two-colored such that each monochromatic connected component is a path of length at most 5 (see Fig. 2(a) for an example). In addition, the aforementioned coloring can be found in polynomial time [22]. Note that in such a coloring of a cubic graph, each vertex appears exactly once as an endpoint of a path, and exactly once as an internal vertex of another path. We next show that this result can be easily extended to graphs with maximum degree at most 3. Lemma 2. The edges of any graph with maximum degree at most 3 can be two-colo... |

9 |
A short proof of the linear arboricity for cubic graphs
- Akiyama, Chvátal
- 1981
(Show Context)
Citation Context ... of cubic graphs, which is a contradiction by [18] unless . It is known that the edges of any cubic graph can be two-colored such that each monochromatic connected component is a path (of any length) =-=[19]-=-. In fact, solving a conjecture of Bermond et al. [20], Thomassen proved [21] a stronger result: The edges of any cubic graph can be two-colored such that each monochromatic connected component is a p... |

8 | Physical Impairment Based Regenerator Placement and Routing
- Pachnicke
- 2008
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6 |
Traffic Independent Heuristics for Regenerator Site Selection for Providing Any-to-Any Optical Connectivity”,
- Saradhi
- 2010
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5 | Static vs. dynamic regenerator assignment in optical switches: models and cost trade-offs.
- Sriram, Griffith, et al.
- 2004
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5 |
Approximation of k -Set Cover by Semi-Local Optimization
- Duh, Fuerer
- 1997
(Show Context)
Citation Context ...placed at a vertex and used by these lightpaths. Clearly, a set cover of minimum cost corresponds to a placement of the minimum number of regenerators -satisfying all the lightpaths. The algorithm of =-=[22]-=- achieves an approximation ratio of ,whereis the maximum size of a set and . In our case, each regenerator covers at most edges of at most lightpaths, so the size of the .MERTZIOS et al.: PLACING REG... |

5 | MPLS label stacking on the line network. In - Bermond, Coudert, et al. - 2009 |

3 |
A Framework for Regenerator Site Selection Based on Multiple Paths.
- Fedrizzi, Galimberti, et al.
- 2010
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2 |
Predeployment of Resources in Agile Photonic Networks.
- Gerstel, Raza
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Citation Context ...r closes a site within the area of the network. In such a situation of uncertain traffic forecast, a common approach in order to minimize capital expenses is to predeploy (or overprovision) resources =-=[12,14,15,17]-=-. That is, the network is designed to satisfy several possible traffic patterns. A similar setting arises in networks in which there are several possible traffic configurations that alternate accordin... |

2 |
Scaling of Most-Likely Traffic Patterns of Hose- and Cost-Constrained Ring and Mesh Networks.
- Korotki, Oikonomou
- 2006
(Show Context)
Citation Context ...r closes a site within the area of the network. In such a situation of uncertain traffic forecast, a common approach in order to minimize capital expenses is to predeploy (or overprovision) resources =-=[12,14,15,17]-=-. That is, the network is designed to satisfy several possible traffic patterns. A similar setting arises in networks in which there are several possible traffic configurations that alternate accordin... |

2 |
Information Theoretic Approach to Traffic Adaptive WDM Networks.
- Murthy, Sinha
- 2005
(Show Context)
Citation Context ...r closes a site within the area of the network. In such a situation of uncertain traffic forecast, a common approach in order to minimize capital expenses is to predeploy (or overprovision) resources =-=[12,14,15,17]-=-. That is, the network is designed to satisfy several possible traffic patterns. A similar setting arises in networks in which there are several possible traffic configurations that alternate accordin... |

2 |
Hardness and approximation of traffic grooming,” Theoret
- Amini, Pérennes, et al.
- 2009
(Show Context)
Citation Context ...exity results when the routing of the requests is not given). We also assume that lightpaths sharing an edge use different wavelengths. That is, we deal with optical networks without traffic grooming =-=[14]-=-. The length of a lightpath is the number of edges it contains. We consider symmetric lightpaths, that is, a lightpath with endpoints and consists of a request from to and a request from to .Theintern... |

1 |
An Overview of the Global Network Expectation Model.
- Korotki
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1 | Griffith,R.Su,andN.Golmie,“Staticvs.dynamicregenerator assignment in optical switches: Models and cost trade-offs - Sriram, D |

1 |
On linear -arboricity
- Bermond, Fouquet, et al.
- 1984
(Show Context)
Citation Context ...ess . It is known that the edges of any cubic graph can be two-colored such that each monochromatic connected component is a path (of any length) [19]. In fact, solving a conjecture of Bermond et al. =-=[20]-=-, Thomassen proved [21] a stronger result: The edges of any cubic graph can be two-colored such that each monochromatic connected component is a path of length at most 5 [see Fig. 2(a) for an example]... |

1 |
tree networks: Complexity, bounds, and algorithms
- Huang, Dutta, et al.
- 2006
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Citation Context ...solution methods for general cases. In this section, we focus on the case where the network topology is a path, which is one of the most studied topologies in theoretical networking (see for instance =-=[23]-=-–[26]), as well as one of the most natural and apparently simplest underlying graphs to study. Our investigations on the path are valuable in several ways. As always, studying elemental cases of a pro... |

1 | Traffic groomingonthe path,” Theoret - Bermond, Braud, et al. - 2007 |

1 | Minimizing transceivers in optical path networks
- Iyer, Dutta, et al.
- 2009
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Citation Context ...ion methods for general cases. In this section, we focus on the case where the network topology is a path, which is one of the most studied topologies in theoretical networking (see for instance [23]–=-=[26]-=-), as well as one of the most natural and apparently simplest underlying graphs to study. Our investigations on the path are valuable in several ways. As always, studying elemental cases of a problem ... |

1 | Brief announcement: On regenerator placement problems in optical networks
- Sen, Banerjee, et al.
(Show Context)
Citation Context ...ned in [9], other interesting avenues for further research are to consider the online setting (that is, when the lightpaths are not given in advance) and the weighted version of the problem (see also =-=[31]-=-), in the sense that the edges of the network have an associated weight, and the distance constraint is replaced with the corresponding weighted distance. Considering the parameterized complexity of t... |