## Minimum Leaf Out-Branching Problems (2008)

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Venue: | Lect. Notes Comput. Sci. 5034 (2008), 235–246 (Proc. AAIM’08 |

Citations: | 7 - 2 self |

### BibTeX

@INPROCEEDINGS{Gutin08minimumleaf,

author = {Gregory Gutin and Igor Razgon and Eun Jung Kim},

title = {Minimum Leaf Out-Branching Problems},

booktitle = {Lect. Notes Comput. Sci. 5034 (2008), 235–246 (Proc. AAIM’08},

year = {2008},

pages = {235--246}

}

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### Abstract

Abstract. Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We describe three parameterizations of MinLOB and prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parametrization is as follows: given a digraph D of order n and a positive integral parameter k, check whether D contains an outbranching with at most n − k leaves (and find such an out-branching if it exists). We find a problem kernel of order O(k · 16 k) and construct an algorithm of running time O(2 O(k log k) + n 2 log n), which is an ‘additive ’ FPT algorithm. 1

### Citations

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Citation Context ... one is fixed-parameter tractable.s2 G. Gutin, I. Razgon, and E.J. Kim We recall some basic notions of parameterized complexity here, for a more in-depth treatment of the topic we refer the reader to =-=[7, 11, 19]-=-. A parameterized problem Π can be considered as a set of pairs (I, k) where I is the problem instance and k (usually an integer) is the parameter. Π is called fixed-parameter tractable (FPT) if membe... |

284 | Invitation to Fixed-Parameter Algorithms
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Citation Context ... one is fixed-parameter tractable.s2 G. Gutin, I. Razgon, and E.J. Kim We recall some basic notions of parameterized complexity here, for a more in-depth treatment of the topic we refer the reader to =-=[7, 11, 19]-=-. A parameterized problem Π can be considered as a set of pairs (I, k) where I is the problem instance and k (usually an integer) is the parameter. Π is called fixed-parameter tractable (FPT) if membe... |

234 | Digraphs: Theory, Algorithms and Applications
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(Show Context)
Citation Context ...]. Notice that not every digraph D has an out-branching. It is not difficult to see that D has an out-branching (i.e., ℓmin(D) > 0) if and only if D has just one strong initial connectivity component =-=[3]-=-. Since the last condition can be checked in linear time [3], we may often assume that ℓmin(D) > 0. Since MinLOB generalizes the hamiltonian directed path problem, MinLOB is NP-hard. In this paper, we... |

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Invitation to data reduction and problem kernelization
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Citation Context ...lem kernels obtained by this general result have impractically large size. Therefore, one tries to develop kernelizations that yield problem kernels of smaller size. The survey of Guo and Niedermeier =-=[12]-=- on kernelization lists some problem for which polynomial size kernels and exponential size kernels were obtained. Notice that a kernelization allows one to obtain so-called additive FPT algorithms, i... |

74 | Treewidth: Computations and Approximations - Kloks - 1994 |

63 |
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Citation Context ...ecomposition of underlying graphs. The parametrization MinLOB-PBGV is of the type below a guaranteed value. Parameterizations above/below a guaranteed value were first considered by Mahajan and Raman =-=[18]-=- for the problems Max-SAT and Max-Cut; such parameterizations have lately gained much attention, cf. [9, 13–15, 19] (it worth noting that Heggernes, Paul, Telle, and Villanger [15] recently solved the... |

45 |
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Citation Context ...time required to perform Step 3 is the same as that of solving the maximum cardinality matching problem on a bipartite graph. The last problem can be solved in time O(|V (B)| 1.5√ |E(B)|/log |V (B)|) =-=[3]-=-. Hence, the algorithm requires at most O(m + n1.5√m/log n) time. ⊓⊔ 3 Parameterizations of MinLOB The following is a natural way to parameterize MinLOB. MinLOB Parameterized Naturally (MinLOB-PN) Ins... |

39 | Dag-width and Parity Games
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Citation Context ...idth have to be modified by 1 taking into consideration that Kelly-width equals elimination-width plus 1). 3s2 Three Directed Decompositions DAG-width was introduced independently by Berwanger et al. =-=[3]-=- and Obdrzalek [8]. A DAG-decomposition (DAGD) of a digraph D is a pair (H, χ) where H is an acyclic digraph and χ = {Wh : h ∈ V (H)} is a family of subsets (called bags) of V (D) satisfying the follo... |

21 | Parameterized Algorithmics: A Graph-Theoretic Approach. Habilitationsschrift. WilhelmSchickard-Institut für Informatik, Universität - Fernau - 2005 |

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Citation Context ...en Wh ∩ Wh ′′ ⊆ Wh ′. The width of a DAGD (H, χ) is max h∈V (H) |Wh| − 1. The DAG-width of a digraph D (dagw(D)) is the minimum width over all possible DAGDs of D. A directed path decomposition (DPD) =-=[2]-=- is a special case of DAGD when H is a directed path. The directed path-width of a digraph D (dpw(D)) is defined as the DAG-width above, but DAGDs are replaced by DPDs. Directed tree-width was introdu... |

16 |
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Citation Context ...dified by 1 taking into consideration that Kelly-width equals elimination-width plus 1). 3s2 Three Directed Decompositions DAG-width was introduced independently by Berwanger et al. [3] and Obdrzalek =-=[8]-=-. A DAG-decomposition (DAGD) of a digraph D is a pair (H, χ) where H is an acyclic digraph and χ = {Wh : h ∈ V (H)} is a family of subsets (called bags) of V (D) satisfying the following three propert... |

13 | Fixed-Parameter Complexity of Minimum Profile Problems - Gutin, Szeider, et al. - 2006 |

12 | The linear arrangement problem parameterized above guaranteed value. Theory Comput - Gutin, Rafiey, et al. |

11 | S.: Parameterized algorithms for directed maximum leaf problems
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Citation Context ...f Out-Branching problem. MaxLOB Parameterized Naturally (MaxLOB-PN) Instance: A digraph D. Parameter: A positive integer k. Question: Does D have an out-branching with at least k leaves ? Alon et al. =-=[1, 2]-=- proved this problem is FPT for several special classes of digraphs such as strongly connected digraphs and acyclic digraphs and Bonsma and Dorn [5] proved that the problem is FPT. Note that in the th... |

10 | S.: Better algorithms and bounds for directed maximum leaf problems
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Citation Context ...f Out-Branching problem. MaxLOB Parameterized Naturally (MaxLOB-PN) Instance: A digraph D. Parameter: A positive integer k. Question: Does D have an out-branching with at least k leaves ? Alon et al. =-=[1, 2]-=- proved this problem is FPT for several special classes of digraphs such as strongly connected digraphs and acyclic digraphs and Bonsma and Dorn [5] proved that the problem is FPT. Note that in the th... |

9 | Interval completion with few edges
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Citation Context ...y Mahajan and Raman [18] for the problems Max-SAT and Max-Cut; such parameterizations have lately gained much attention, cf. [9, 13–15, 19] (it worth noting that Heggernes, Paul, Telle, and Villanger =-=[15]-=- recently solved the longstanding minimum interval completion problem, which is a parametrization above guaranteed value). For directed graphs there have been only a couple of results on problems para... |

8 |
Digraph Measures: Kelly Decompositions
- Hunter, Kreutzer
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Citation Context ...path-width, directed tree-width and DAG-width as they appear to be the most studied directed width parameters, but our results hold for other width parameters such as elimination width and Kellywidth =-=[6]-=- (the results for Kelly-width have to be modified by 1 taking into consideration that Kelly-width equals elimination-width plus 1). 3s2 Three Directed Decompositions DAG-width was introduced independe... |

7 | Parameterized algorithmics for linear arrangement problems
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Citation Context ...rval completion problem, which is a parametrization above guaranteed value). For directed graphs there have been only a couple of results on problems parameterized above/below a guaranteed value, see =-=[4, 10]-=-. In particular, Bang-Jensen and Yeo [4] proved that the following problem is FPT. Let ms(D) denote the minimum number of arcs in a strongly connected spanning subgraph of D. Minimum Spanning Strong S... |

5 | An FPT algorithm for directed spanning k-leaf
- Bonsma, Dorn
- 2007
(Show Context)
Citation Context ...t-branching with at least k leaves ? Alon et al. [1, 2] proved this problem is FPT for several special classes of digraphs such as strongly connected digraphs and acyclic digraphs and Bonsma and Dorn =-=[5]-=- proved that the problem is FPT. Note that in the three papers, MaxLOB-PN algorithms are of running time O(2k(log k)O(1) · nO(1) ). Acknowledgements We are grateful to Bruno Courcelle for useful discu... |

5 | Algorithms for Enumerating all Spanning Trees of Directed and Undirected Graphs
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(Show Context)
Citation Context ... · 16 k . Now we have to clarify how we explore this kernel in order to get the desired out-branching. A straightforward exploration of all possible out-branchings (using, e.g., the main algorithm of =-=[16]-=-) is not a good choice because the number of different out-branchings may be up to p p−1 , where p = |V (D ′ )| = (k · 16 k ). Indeed, by the famous Kelly’s formula the number of spanning trees in the... |

4 |
Minimum leaf spanning tree
- Demers, Downing
- 2008
(Show Context)
Citation Context ...nimum possible number of leaves. Denote this minimum by ℓmin(D). When D has no out-branching, we write ℓmin(D) = 0. The MinLOB problem has applications in the area of database systems, cf. the patent =-=[8]-=-. Notice that not every digraph D has an out-branching. It is not difficult to see that D has an out-branching (i.e., ℓmin(D) > 0) if and only if D has just one strong initial connectivity component [... |

2 |
The minimum spanning strong subdigraph problem is fixed parameter tractable
- Bang-Jensen, Yeo
(Show Context)
Citation Context ...rval completion problem, which is a parametrization above guaranteed value). For directed graphs there have been only a couple of results on problems parameterized above/below a guaranteed value, see =-=[4, 10]-=-. In particular, Bang-Jensen and Yeo [4] proved that the following problem is FPT. Let ms(D) denote the minimum number of arcs in a strongly connected spanning subgraph of D. Minimum Spanning Strong S... |