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## Exact algorithms and applications for Tree-like Weighted Set Cover (2006)

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Venue: | JOURNAL OF DISCRETE ALGORITHMS |

Citations: | 11 - 6 self |

### Citations

13825 |
Computers and Intractability - A Guide to the Theory of NPCompleteness, Bell Telephone Laboratories, Incorporated
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...he number of graph vertices and parameter k denotes the size of the vertex cover set asked for. 2.1 Tree-Like Weighted Set Cover (Weighted) Set Cover is one of the most prominent NP-complete problems =-=[14]-=-. The basic Set Cover problem (optimization version) is defined as follows: Input: A base set S = {s1, s2, . . .,sn} and a collection C of subsets of S, C = {c1, c2, . . .,cm}, ci ⊆ S for 1 ≤ i ≤ m, a... |

1365 |
Integer and Combinatorial Optimization
- Nemhauser, Wolsey
- 1988
(Show Context)
Citation Context ...ypergraph cyclicity and how to make use of “bounded cyclicity”. The special case of TWSC where T should only be a path has been studied under the name Set Cover with Consecutive Ones Property (SCC1P) =-=[19,22]-=-. A Set Cover instance can also be represented by a binary coefficient matrix M over {0, 1} where the rows correspond to the elements in S, the columns correspond to the subsets in C, and an entry is ... |

767 | A Threshold of ln n for Approximating Set Cover
- Feige
- 1998
(Show Context)
Citation Context ...pears to be very hard from an algorithmic point of view. Unless NP has slightly super-polynomial time algorithms, the best polynomial-time approximation algorithm achieves approximation factor Θ(lnn) =-=[11]-=-, where n denotes the base set size. In addition, from the viewpoint of parameterized complexity, the problem is known to be W[2]-complete [9] with respect to the parameter “number of chosen covering ... |

605 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ... s ∈ S in C as the number of the subsets in C which contain s. An element with occurrence of one is called unique. Set Cover remains NP-complete even if the occurrence of each element is bounded by 2 =-=[26]-=-. Definition 1 (Tree-like subset collection) A base set S = {s1, s2, . . .,sn} and a collection C of subsets of S, C = {c1, c2, . . .,cm}, ci ⊆ S for 1 ≤ i ≤ m. We say that C is a tree-like subset col... |

433 | Invitation to Fixed-Parameter Algorithms
- Niedermeier
- 2006
(Show Context)
Citation Context ...ubmitted to Elsevier Preprint 16 June 2005sJournal of Discrete Algorithms, Vol. 4(4), pp. 608-622, 2006 with fixed-parameter algorithms that emerge from the field of parameterized complexity analysis =-=[9,24]-=-. The point here is to detect relevant (and hopefully small) parameters in instances of hard problems and to try to restrict the seemingly unavoidable combinatorial explosions in the running times of ... |

432 |
Graph minors { II. Algorithmic aspects of tree-width
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...lar, “tree-like” refers to the fact that the subset collection has to fulfill an acyclicity property (called consistency property later) known from the famous concept of tree decompositions of graphs =-=[27,20,4,5]-=-. It is not very hard to see that a subset collection fulfills the “tree-like property” iff it can be interpreted as an acyclic hypergraph as studied in a somewhat different context by Tarjan and Yann... |

317 | A partial k-arboretum of graphs with bounded treewidth
- BODLAENDER
- 1998
(Show Context)
Citation Context ...lar, “tree-like” refers to the fact that the subset collection has to fulfill an acyclicity property (called consistency property later) known from the famous concept of tree decompositions of graphs =-=[27,20,4,5]-=-. It is not very hard to see that a subset collection fulfills the “tree-like property” iff it can be interpreted as an acyclic hypergraph as studied in a somewhat different context by Tarjan and Yann... |

298 |
Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs
- Tarjan, Yannakakis
- 1984
(Show Context)
Citation Context ... not very hard to see that a subset collection fulfills the “tree-like property” iff it can be interpreted as an acyclic hypergraph as studied in a somewhat different context by Tarjan and Yannakakis =-=[29]-=-. Hence, the Tree-like Weighted Set Cover problem is equivalent to the Minimum Weighted Edge Cover problem on acyclic hypergraphs. The tree-likeness property of a set system can be checked in linear t... |

150 | Treewidth: Algorithmic techniques and results, in: The Twenty
- Bodlaender
- 1997
(Show Context)
Citation Context ...lar, “tree-like” refers to the fact that the subset collection has to fulfill an acyclicity property (called consistency property later) known from the famous concept of tree decompositions of graphs =-=[27,20,4,5]-=-. It is not very hard to see that a subset collection fulfills the “tree-like property” iff it can be interpreted as an acyclic hypergraph as studied in a somewhat different context by Tarjan and Yann... |

146 |
Primal-dual approximation algorithms for integral flow and multicuts in trees
- Garg, Yannakakis, et al.
- 1997
(Show Context)
Citation Context ...rameter tractability result for TWSC, we can also show that the so-called Multicut in Trees problem (also known as Weighted Set Cover with tree-representable set system and being different from TWSC) =-=[15]-=- is solvable in O(m · n · 3 k ) time, where m denotes the number of node pairs, n denotes the number of tree nodes, and k is the maximum number of paths passing through some tree node. Multicut in Tre... |

111 | Fixed parameter algorithms for dominating set and related problems on planar graphs
- Alber, Bodlaender, et al.
(Show Context)
Citation Context ... of a hypergraph. 3sJournal of Discrete Algorithms, Vol. 4(4), pp. 608-622, 2006 that our basic technique in a sense is similar in spirit to dynamic programming on tree decompositions as exhibited in =-=[30,1]-=-. We emphasize that the trivial solution for (Tree-like) Weighted Set Cover (not considering parameter k) has running time (ignoring polynomial factors) O(2 m ), where m = O(2 n ) is possible. By way ... |

54 |
Treewidth: computations and approximations, volume 842 of LNCS
- Kloks
- 1994
(Show Context)
Citation Context |

40 | Parameterized complexity for the skeptic
- Downey
- 2003
(Show Context)
Citation Context ...losions in the running times of the solving algorithms to these parameters. For small parameter values—as occur in many problem instances of practical interest—this leads to efficient algorithms; see =-=[8,12,13,23]-=- for surveys. Here we report on progress in dealing with a practically relevant special case of the Weighted Set Cover problem. Set Cover is one of the most important problems in combinatorial optimiz... |

40 | Approximation of k-set cover by semi-local optimization - Duh, Fürer - 1997 |

39 | Hypertree decompositions: A survey - Gottlob, Leone, et al. - 2001 |

32 | Vertebrate phylogenomics: reconciled trees and gene
- Page, Cotton
- 2002
(Show Context)
Citation Context ...formalization, experiments showed memory savings of around 80 to 90 % [2]. The second application arises in computational molecular biology. In their work on vertebrate phylogenomics, Page and Cotton =-=[25]-=- formulate the problem of locating gene duplications as Tree-like Unweighted Set Cover: given a so-called species tree in which each node has a set of gene duplications associated with it, find the sm... |

31 |
An Algorithm for Determining Whether a Given Binary Matroid is Graphic
- Tutte
- 1960
(Show Context)
Citation Context ...ecide whether a given Set Cover instance is tree-representable [3]. The problem of deciding whether a given Set Cover instance is tree-representable has been extensively studied in different contexts =-=[31]-=-. Compare Tree-representable Set Cover with Tree-like Set Cover. Both problems have an underlying tree and, in both problems, the subsets containing an element should induce a connected substructure. ... |

28 | Approximating k-set cover and complementary graph coloring
- Halld'orsson
- 1996
(Show Context)
Citation Context ...countered parameter values smaller than 50. Note that the parameter “maximum subset size” also played an important role in developing polynomial-time approximation algorithms for unweighted Set Cover =-=[10,18]-=-, after a series of papers the best approximation factor for subset size 3 now being 4/3 [10]. It is an open question of Halldórsson [18], however, to obtain similar results for the weighted version o... |

22 |
Optimal capacity scheduling
- Veinott, Wagner
- 1962
(Show Context)
Citation Context ...ypergraph cyclicity and how to make use of “bounded cyclicity”. The special case of TWSC where T should only be a path has been studied under the name Set Cover with Consecutive Ones Property (SCC1P) =-=[19,22]-=-. A Set Cover instance can also be represented by a binary coefficient matrix M over {0, 1} where the rows correspond to the elements in S, the columns correspond to the subsets in C, and an entry is ... |

22 |
Practical algorithms on partial k-trees with an application to domination-like problems
- Telle, Proskurowski
- 1998
(Show Context)
Citation Context ... of a hypergraph. 3sJournal of Discrete Algorithms, Vol. 4(4), pp. 608-622, 2006 that our basic technique in a sense is similar in spirit to dynamic programming on tree decompositions as exhibited in =-=[30,1]-=-. We emphasize that the trivial solution for (Tree-like) Weighted Set Cover (not considering parameter k) has running time (ignoring polynomial factors) O(2 m ), where m = O(2 n ) is possible. By way ... |

20 | Simplicity is Beauty: Improved Upper Bounds for Vertex Cover
- Chen, Kanjy, et al.
- 2005
(Show Context)
Citation Context ... depending on k (see [9,8,12,13,23,24] for more details). For instance, the best known fixed-parameter algorithm for the Vertex Cover problem is known to run in less than O(kn+1.3 k ) worst-case time =-=[6,7]-=-, 4sJournal of Discrete Algorithms, Vol. 4(4), pp. 608-622, 2006 where n denotes the number of graph vertices and parameter k denotes the size of the vertex cover set asked for. 2.1 Tree-Like Weighted... |

20 |
New directions and new challenges in algorithm design and complexity, parameterized
- Fellows
- 2003
(Show Context)
Citation Context ...losions in the running times of the solving algorithms to these parameters. For small parameter values—as occur in many problem instances of practical interest—this leads to efficient algorithms; see =-=[8,12,13,23]-=- for surveys. Here we report on progress in dealing with a practically relevant special case of the Weighted Set Cover problem. Set Cover is one of the most important problems in combinatorial optimiz... |

20 |
crown rules: Some new directions in FPT
- Blow-ups
- 2003
(Show Context)
Citation Context ...losions in the running times of the solving algorithms to these parameters. For small parameter values—as occur in many problem instances of practical interest—this leads to efficient algorithms; see =-=[8,12,13,23]-=- for surveys. Here we report on progress in dealing with a practically relevant special case of the Weighted Set Cover problem. Set Cover is one of the most important problems in combinatorial optimiz... |

17 | Solving geometric covering problems by data reduction
- Mecke, Wagner
- 2004
(Show Context)
Citation Context ... parameterization. Using TWSC, memory savings of up to 90 % and more have been achieved in this way [2]. An interesting potential for further applications appears in a recent work of Mecke and Wagner =-=[21]-=- in the context of railway optimization problems. Other applications of TWSC are conceivable with respect to Multicut in Trees [15], acyclic hypergraphs and their applications for relational databases... |

14 | R.: Fixed-parameter tractability and data reduction for multicut in trees
- Guo, Niedermeier
- 2005
(Show Context)
Citation Context ...icut in Trees instance and vice versa. ✷ We remark that in a related work we have shown that unweighted Multicut in Trees is fixed-parameter tractable with respect to the size of the solution set E ′ =-=[17]-=-. Garg et al. [15] have shown that Multicut in Trees is equivalent to the so-called Tree-representable Set Cover problem. A (weighted) Set Cover instance (S, C) is called a tree-representable set syst... |

12 | Ubiquitous parameterization—invitation to fixed-parameter algorithms
- Niedermeier
(Show Context)
Citation Context |

12 | Set covering with almost consecutive ones property
- Ruf, Schöbel
(Show Context)
Citation Context ...nally, it is an interesting task for future research to investigate the relation between tree-like set covering and set covering with almost consecutive ones property as introduced by Ruf and Schöbel =-=[28]-=-. 5 The corresponding parameters are the maximum subset size and the maximum number of paths passing through a node or an edge, respectively. See [17] for a fixed-parameter tractability result for Mul... |

8 | Tree decompositions of graphs: Saving memory in dynamic programming
- Betzler, Niedermeier, et al.
(Show Context)
Citation Context ...ial-time approximation algorithm achieves approximation factor Θ(lnn) [11], where n denotes the base set size. In addition, from the viewpoint of parameterized complexity, the problem is known to be W=-=[2]-=--complete [9] with respect to the parameter “number of chosen covering sets” which excludes fixed-parameter tractability in this respect. Hence, a prospective way out of this misery seems to be to fur... |

8 |
An almost linear time algorithm for graph realization
- Bixby, Wagner
- 1985
(Show Context)
Citation Context ...each element s in S, the edges corresponding to the subsets containing s induce a path in T. There are almost linear-time algorithms to decide whether a given Set Cover instance is tree-representable =-=[3]-=-. The problem of deciding whether a given Set Cover instance is tree-representable has been extensively studied in different contexts [31]. Compare Tree-representable Set Cover with Tree-like Set Cove... |

6 | Refined memorisation for vertex cover
- Chandran, Grandoni
- 2004
(Show Context)
Citation Context ... depending on k (see [9,8,12,13,23,24] for more details). For instance, the best known fixed-parameter algorithm for the Vertex Cover problem is known to run in less than O(kn+1.3 k ) worst-case time =-=[6,7]-=-, 4sJournal of Discrete Algorithms, Vol. 4(4), pp. 608-622, 2006 where n denotes the number of graph vertices and parameter k denotes the size of the vertex cover set asked for. 2.1 Tree-Like Weighted... |