## Parameterized complexity of generalized vertex cover problems (2005)

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Venue: | In Proc. 9th WADS, volume 3608 of LNCS |

Citations: | 19 - 2 self |

### BibTeX

@INPROCEEDINGS{Guo05parameterizedcomplexity,

author = {Jiong Guo and Rolf Niedermeier and Sebastian Wernicke},

title = {Parameterized complexity of generalized vertex cover problems},

booktitle = {In Proc. 9th WADS, volume 3608 of LNCS},

year = {2005},

pages = {36--48},

publisher = {Springer}

}

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### Abstract

Abstract. Important generalizations of the Vertex Cover problem

### Citations

867 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ... Cover problem is to find a set C ⊆ V with |C| ≤ k such that each edge in E has at least one endpoint in C. In a sense, Vertex Cover could be considered the Drosophila of fixed-parameter algorithmics =-=[17, 25]-=-: 1. There is a long list of continuous improvements on the combinatorial explosion with respect to the parameter k when solving the problem exactly. The currently best exponential bound is below 1.28... |

319 |
Introduction to Algorithms, 2 nd edition
- Cormen, Leiserson, et al.
- 2001
(Show Context)
Citation Context ...n edge in E ′ if and only if the vertex represented by u matches the profile represented by v (the weight of the edge is w(u)). Finding such a matching is well-known to be solvable in polynomial time =-=[16]-=-. ⊓⊔ It is possible to solve Soft CVC and Hard CVC by adapting the above algorithm for CVC: Observe that if we choose multiple copies of a vertex into the cover, each of these copies will have its own... |

284 | Invitation to Fixed-Parameter Algorithms
- Niedermeier
- 2006
(Show Context)
Citation Context ... Cover problem is to find a set C ⊆ V with |C| ≤ k such that each edge in E has at least one endpoint in C. In a sense, Vertex Cover could be considered the Drosophila of fixed-parameter algorithmics =-=[17, 25]-=-: 1. There is a long list of continuous improvements on the combinatorial explosion with respect to the parameter k when solving the problem exactly. The currently best exponential bound is below 1.28... |

150 | Vertex cover: further observations and further improvements
- Chen, Kanj, et al.
- 1999
(Show Context)
Citation Context .... There is a long list of continuous improvements on the combinatorial explosion with respect to the parameter k when solving the problem exactly. The currently best exponential bound is below 1.28 k =-=[8, 26, 14, 28, 12]-=-. 2. Vertex Cover has been a benchmark for developing sophisticated data reduction and problem kernelization techniques [1, 19]. 3. It was the first parameterized problem where the usefulness of inter... |

103 | Fixed parameter algorithms for dominating set and related problems on planar graphs, Algorithmica 33 - Alber, Bodlaender, et al. - 2002 |

92 | Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs
- Halperin
(Show Context)
Citation Context ...Guha et al. [22] who also give a factor-2 approximation algorithm. Two special flavors of CVC exist in the literature that arise by allowing “copies” of a vertex to be in the capacitated vertex cover =-=[22, 15, 20]-=-. In that context, taking a vertex l times into the capacitated vertex cover causes the vertex to have l times its original capacity. The number of such copies is unlimited in the Soft Capacitated Ver... |

73 | The Steiner problem in graphs - Dreyfus, Wagner - 1972 |

60 | Parameterized complexity: exponential speed-up for planar graph problems
- Alber, Fernau, et al.
- 2001
(Show Context)
Citation Context ...ed to planar graphs, it was—besides Dominating Set—one of the central problems for the development of “subexponential” fixed-parameter algorithms and the corresponding theory of relative lower bounds =-=[2, 4, 11]-=-. 5. Vertex Cover served as a testbed for algorithm engineering in the realm of fixed-parameter algorithms [1, 3, 13]. ⋆ Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group P... |

45 | A general method to speed up fixed-parametertractable algorithms
- Niedermeier, Rossmanith
- 2000
(Show Context)
Citation Context ... reduction and problem kernelization techniques [1, 19]. 3. It was the first parameterized problem where the usefulness of interleaving depth-bounded search trees and problem kernelization was proven =-=[27]-=-. 4. Restricted to planar graphs, it was—besides Dominating Set—one of the central problems for the development of “subexponential” fixed-parameter algorithms and the corresponding theory of relative ... |

44 | Approximation algorithms for partial covering problems
- Gandhi, Khuller, et al.
(Show Context)
Citation Context ... ⊆ V of size at most k such that V ′ covers at least t edges. This problem was introduced by Bshouty and Burroughs [10] who showed it to be approximable within 2. Further improvements can be found in =-=[21]-=-. Note that Maximum Partial Vertex Cover is fixed-parameter tractable with respect to the parameter t [9]. In case of Minimum Partial Vertex Cover we are asked for a vertex subset with at least k vert... |

43 | Upper bounds for Vertex Cover further improved
- Niedermeier, Rossmanith
- 1999
(Show Context)
Citation Context .... There is a long list of continuous improvements on the combinatorial explosion with respect to the parameter k when solving the problem exactly. The currently best exponential bound is below 1.28 k =-=[8, 26, 14, 28, 12]-=-. 2. Vertex Cover has been a benchmark for developing sophisticated data reduction and problem kernelization techniques [1, 19]. 3. It was the first parameterized problem where the usefulness of inter... |

41 | Kernelization algorithms for the vertex cover problem: Theory and experiments
- Abu-Khzam, Collins, et al.
- 2004
(Show Context)
Citation Context ... by showing that, with the size of the desired vertex cover as parameter, Connected Vertex Cover and Capacitated Vertex Cover are both fixed-parameter tractable while Maximum Partial Vertex Cover is W=-=[1]-=--hard. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered generalized Vertex Cover problems behave very... |

36 | On the existence of subexponential parameterized algorithms
- Cai, Juedes
- 2003
(Show Context)
Citation Context ...ed to planar graphs, it was—besides Dominating Set—one of the central problems for the development of “subexponential” fixed-parameter algorithms and the corresponding theory of relative lower bounds =-=[2, 4, 11]-=-. 5. Vertex Cover served as a testbed for algorithm engineering in the realm of fixed-parameter algorithms [1, 3, 13]. ⋆ Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group P... |

32 | Covering problems with hard capacities
- Chuzhoy, Naor
- 2006
(Show Context)
Citation Context ...Guha et al. [22] who also give a factor-2 approximation algorithm. Two special flavors of CVC exist in the literature that arise by allowing “copies” of a vertex to be in the capacitated vertex cover =-=[22, 15, 20]-=-. In that context, taking a vertex l times into the capacitated vertex cover causes the vertex to have l times its original capacity. The number of such copies is unlimited in the Soft Capacitated Ver... |

31 | Faster exact algorithms for hard problems: a parameterized point of view, Discrete Math
- Alber, Gramm, et al.
- 2001
(Show Context)
Citation Context ... one only wants to cover a specified number of edges (that is, not necessarily all) by at most k vertices. This problem is known to have a factor-2 approximation [10]. Answering an open question from =-=[5]-=-, we show that this problem appears to be fixed-parameter intractable—it is W[1]-hard. The same is proven for its minimization version.sProc. 9th WADS-05, Vol. 3608 in LNCS, pp. 36-48, Springer, 2005 ... |

31 | On efficient fixed-parameter algorithms for weighted vertex cover
- Niedermeier, Rossmanith
(Show Context)
Citation Context .... There is a long list of continuous improvements on the combinatorial explosion with respect to the parameter k when solving the problem exactly. The currently best exponential bound is below 1.28 k =-=[8, 26, 14, 28, 12]-=-. 2. Vertex Cover has been a benchmark for developing sophisticated data reduction and problem kernelization techniques [1, 19]. 3. It was the first parameterized problem where the usefulness of inter... |

29 |
An Improved Fixed-Parameter Algorithm for Vertex Cover
- Balasubramanian, Fellows, et al.
(Show Context)
Citation Context |

19 |
New directions and new challenges in algorithm design and complexity, parameterized
- Fellows
(Show Context)
Citation Context ...xactly. The currently best exponential bound is below 1.28 k [8, 26, 14, 28, 12]. 2. Vertex Cover has been a benchmark for developing sophisticated data reduction and problem kernelization techniques =-=[1, 19]-=-. 3. It was the first parameterized problem where the usefulness of interleaving depth-bounded search trees and problem kernelization was proven [27]. 4. Restricted to planar graphs, it was—besides Do... |

18 |
Approximation algorithms for some parameterized counting problems
- Arvind, Raman
(Show Context)
Citation Context ...iftung des deutschen Volkes.sProc. 9th WADS-05, Vol. 3608 in LNCS, pp. 36-48, Springer, 2005 6. Studies of Vertex Cover led to new research directions within parameterized complexity such as counting =-=[7]-=-, parallel processing [13], or using “vertex cover structure” as a general strategy to solve parameterized problems [30]. This probably incomplete list gives an impression of how important Vertex Cove... |

17 | Approximating the tree and tour covers of a graph
- Arkin, Halldórsson, et al.
- 1993
(Show Context)
Citation Context .... The parameter k always denotes the size of the vertex cover. 1. For Connected Vertex Cover one demands that the vertex cover set is connected. This problem is known to have a factor-2 approximation =-=[6]-=-. We show that it can be solved in O(6 k n + 4 k n 2 + 2 k n 2 log n + 2 k nm) time. In addition, we derive results for the closely related variants Tree Cover and Tour Cover. 2. For Capacitated Verte... |

16 | Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem
- Bshouty, Burroughs
- 1998
(Show Context)
Citation Context .... 3. For Maximum Partial Vertex Cover, one only wants to cover a specified number of edges (that is, not necessarily all) by at most k vertices. This problem is known to have a factor-2 approximation =-=[10]-=-. Answering an open question from [5], we show that this problem appears to be fixed-parameter intractable—it is W[1]-hard. The same is proven for its minimization version.sProc. 9th WADS-05, Vol. 360... |

16 | Capacitated vertex covering
- Guha, Hassin, et al.
- 2003
(Show Context)
Citation Context ...is field to date appears to have neglected a closer investigation of recent significant generalizations and variants of Vertex Cover. These appear in various application scenarios such as drug design =-=[22]-=- and have so far only been studied in the context of their polynomial-time approximability. We close this gap here by providing several first-time parameterized complexity results, which also answers ... |

13 | Experimental evaluation of a tree decomposition based algorithm for vertex cover on planar graphs
- Alber, Dorn, et al.
(Show Context)
Citation Context ...” fixed-parameter algorithms and the corresponding theory of relative lower bounds [2, 4, 11]. 5. Vertex Cover served as a testbed for algorithm engineering in the realm of fixed-parameter algorithms =-=[1, 3, 13]-=-. ⋆ Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group PIAF (fixed-parameter algorithms), NI 369/4. ⋆⋆ Supported by Deutsche Telekom Stiftung and Studienstiftung des deutsch... |

13 | Ubiquitous parameterization—invitation to fixed-parameter algorithms
- Niedermeier
- 2004
(Show Context)
Citation Context ...exity class for fixed-parameter intractability is W[1] as there is good reason to believe that W[1]-hard problems are not fixed-parameter tractable [17]. 1 For a more detailed introduction see, e.g., =-=[17, 19, 24]-=-.sProc. 9th WADS-05, Vol. 3608 in LNCS, pp. 36-48, Springer, 2005 In this work, we consider three directions of generalizing Vertex Cover (VC), namely demanding that the vertices of the cover must be ... |

12 | Thilikos, Fast fixedparameter tractable algorithms for nontrivial generalizations of vertex cover
- Nishimura, Ragde, et al.
(Show Context)
Citation Context ...ameterized complexity results, which also answers two open questions from the literature. We are only aware of two papers that perform somewhat related research. First, Nishimura, Ragde, and Thilikos =-=[29]-=- also study generalizations of Vertex Cover. However, they follow a completely different route: Whereas we study concrete problems such as Capacitated Vertex Cover or Maximum Partial Vertex Cover on g... |

11 | Solving large FPT problems on coarse-grained parallel machines
- Cheetham, Dehne, et al.
- 2003
(Show Context)
Citation Context ...” fixed-parameter algorithms and the corresponding theory of relative lower bounds [2, 4, 11]. 5. Vertex Cover served as a testbed for algorithm engineering in the realm of fixed-parameter algorithms =-=[1, 3, 13]-=-. ⋆ Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group PIAF (fixed-parameter algorithms), NI 369/4. ⋆⋆ Supported by Deutsche Telekom Stiftung and Studienstiftung des deutsch... |

11 | C.: Either/or: using vertex cover structure in designing FPTalgorithms—the case of k-Internal SpanningTree
- Prieto, Sloper
- 2003
(Show Context)
Citation Context ... led to new research directions within parameterized complexity such as counting [7], parallel processing [13], or using “vertex cover structure” as a general strategy to solve parameterized problems =-=[30]-=-. This probably incomplete list gives an impression of how important Vertex Cover was and continues to be for the whole field of parameterized complexity. However, research in this field to date appea... |

10 |
Computing small partial coverings
- Bläser
(Show Context)
Citation Context ...aphs, their interest lies in recognizing general classes of graphs with a very special case of interest being the class of graphs with bounded vertex cover (refer to [29] for details). Second, Bläser =-=[9]-=- shows that some partial covering problems are fixed-parameter tractable when the parameter is the number of objects covered instead of the size of the covering set. (In this paper, as well as in the ... |

6 | Refined memorisation for vertex cover
- Chandran, Grandoni
- 2004
(Show Context)
Citation Context |

6 | Improved approximations for tour and tree covers
- Könemann, Konjevod, et al.
(Show Context)
Citation Context ...can contain repeated vertices and edges. Both Tree Cover and Tour Cover were introduced in [6] where it is shown that Tree Cover is approximable within 3.55 and Tour Cover within 5.5. Könemann et al. =-=[23]-=- improved both approximation factors to 3. Section 4 considers the Capacitated Vertex Cover (CVC) problem and related variants. Here, each vertex v ∈ V is assigned a capacity c(v) ∈ N + that limits th... |