## Combinatorial Algorithms

### BibTeX

@MISC{Drescher_combinatorialalgorithms,

author = {Matthew Drescher},

title = {Combinatorial Algorithms},

year = {}

}

### OpenURL

### Abstract

We present an O ( √ opt)-approximation algorithm for the maximum leaf spanning arborescence problem, where opt is the number of leaves in an optimal spanning arborescence. The result is based upon an O(1)-approximation algorithm for a special class of directed graphs called willows. Incorporating the method for willow graphs as a subroutine in a local improvement algorithm gives the bound for general directed graphs.

### Citations

284 | Approximation algorithms for connected dominating sets
- Guha, Khuller
- 1998
(Show Context)
Citation Context ... a spanning tree T in a undirected graph G containing the maximum number of leaves. This is know as the Maximum Leaf Spanning Tree (mlst) problem and has applications in communication network design [=-=Guha and Khuller 1996-=-], circuit layouts [Storer 1981], and distributed systems [Payan et al. 1984]. This problem, however, is hard; Galbiati et al [Galbiati et al. 2004] proved it to be maxsnp-complete. Given this, there ... |

26 | Approximating Maximum Leaf Spanning Trees in Almost Linear Time
- Lu, Ravi
- 1998
(Show Context)
Citation Context ...earch is then repeated until no improved tree can be found. Lu and Ravi proved that this produced approximation guarantees of 5 and 3, respectively, for the cases k = 1 and k = 2. Later, Lu and Ravi [=-=Lu and Ravi 1998-=-] proved that greedy algorithms also perform well. They presented a nearly linear time algorithm with a factor 3 guarantee. The basic approach is to greedily grow a forest (according to specific expan... |

26 | 2-approximation algorithm for finding a spanning tree with the maximum number of leaves
- Solis-oba
- 1998
(Show Context)
Citation Context ...ith many leaves. Using the structural properties of the resulting forest, they show that the forest can then be extended to a spanning tree with a relatively small cost in terms of leaves. Solis-Oba [=-=Solis-Oba 1998-=-] subsequently refined this approach to produce a factor 2 approximation algorithm. In this paper we consider directed graphs. The analog of spanning tree in a directed graph is called an arborescence... |

20 |
A short note on the approximability of the maximum leaves spanning tree problem
- Galbiati, Maffioli, et al.
- 1994
(Show Context)
Citation Context ...d has applications in communication network design [Guha and Khuller 1996], circuit layouts [Storer 1981], and distributed systems [Payan et al. 1984]. This problem, however, is hard; Galbiati et al [=-=Galbiati et al. 2004-=-] proved it to be maxsnp-complete. Given this, there has been much work on designing approximation algorithms. Interestingly, it turns out that the mlst problem is one of those select problems for whi... |

14 |
Constructing full spanning trees for cubic graphs
- Storer
- 1981
(Show Context)
Citation Context ...G containing the maximum number of leaves. This is know as the Maximum Leaf Spanning Tree (mlst) problem and has applications in communication network design [Guha and Khuller 1996], circuit layouts [=-=Storer 1981-=-], and distributed systems [Payan et al. 1984]. This problem, however, is hard; Galbiati et al [Galbiati et al. 2004] proved it to be maxsnp-complete. Given this, there has been much work on designing... |

11 | S.: Parameterized algorithms for directed maximum leaf problems
- Alon, Fomin, et al.
(Show Context)
Citation Context ...reedy algorithm. For example, greedily growing a forest will fail as the arc directions may then prohibit the forest from being connected up efficiently. 1.1 Related Work. Very recently, Alon et al. [=-=Alon et al. 2007-=-a], [Alon et al. 2007b] examined the parameterised complexity of mlsa. They showed that mlsa is fixed parameter tractable for a large class of directed graphs. Specifically, their algorithm applies to... |

10 | S.: Better algorithms and bounds for directed maximum leaf problems - Alon, Fomin, et al. |

8 |
Approximation hardness of dominating set problems
- Chlebík, Chlebíková
(Show Context)
Citation Context ...algorithm is an O( √ opt)-approximation algorithm. 4. HARDNESS The mlst problem, and thus the mlsa problem, is maxsnp-complete [Galbiati et al. 2004] and is np-hard to approximate within 1 + (1/244) [=-=Chlebik and Chlebikova 2004-=-]. Clearly, there is a huge gap between this lower bound and our upper bound and closing this gap is the key open problem here. For the weighted versions of mlst and mlsa, good approximation algorithm... |

2 |
Arbres avec un Nombre de Maximum de Sommets Pendants
- Payan, Tchuente, et al.
- 1984
(Show Context)
Citation Context ...es. This is know as the Maximum Leaf Spanning Tree (mlst) problem and has applications in communication network design [Guha and Khuller 1996], circuit layouts [Storer 1981], and distributed systems [=-=Payan et al. 1984-=-]. This problem, however, is hard; Galbiati et al [Galbiati et al. 2004] proved it to be maxsnp-complete. Given this, there has been much work on designing approximation algorithms. Interestingly, it ... |

1 |
An fpt algorithm for directed spanning k-leaf. preprint
- Bonsma, Dorn
- 2007
(Show Context)
Citation Context ...h D ∈ F has a spanning arborescence with at least k leaves. We remark that they left as an open problem the parameterized complexity of mlsa for all digraphs, which was even more recently settled in [=-=Bonsma and Dorn 2007-=-]. Although the problem of determining the parameterized complexity and the probJournal of the ACM, Vol. V, No. N, Month 20YY.An Approximation Algorithm for the Maximum Leaf Spanning Arborescence Pro... |

1 |
The power of local optimzation: Approximation algorithms for maximum-leaf spanning tree
- Lu, Ravi
- 1992
(Show Context)
Citation Context ...ar, Lu and Ravi have shown that both local improvement and greedy techniques can be applied successfully. First, they gave constant-factor approximation algorithm using a local improvement algorithm [=-=Lu and Ravi 1992-=-]. Specifically, they analysed a k-exchange algorithm. This involves taking a spanning tree T and exhaustively searching for a spanning tree T ′ that differs from T in at most k edges ∗ School of Comp... |