## Approximation Algorithms for Connected Dominating Sets (1996)

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Venue: | Algorithmica |

Citations: | 296 - 9 self |

### BibTeX

@ARTICLE{Guha96approximationalgorithms,

author = {Sudipto Guha and Samir Khuller},

title = {Approximation Algorithms for Connected Dominating Sets},

journal = {Algorithmica},

year = {1996},

volume = {20},

pages = {374--387}

}

### Years of Citing Articles

### OpenURL

### Abstract

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...

### Citations

11575 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...inating set problem is defined as follows. Find a minimum size subset S of vertices, such that the subgraph induced by S is connected and S forms a dominating set. This problem is known to be NP-hard =-=[7]-=-. Recall that a dominating set is one in which each vertex is either in the dominating set, or adjacent to some vertex in the dominating set. A preliminary version of this paper will appear in the Pro... |

9130 | Introduction to algorithms - Cormen, Leiserson, et al. - 2001 |

394 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1992
(Show Context)
Citation Context ...approximation preserving reduction from the set-cover problem to the connected dominating set problem, showing that it is hard to improve the approximation guarantee unless NP ` DT IME[n O(loglogn) ] =-=[13, 6]-=-. We give a 3 ln n approximation for the version when the vertices have weights. We also show that the upper bound of 2 ln k for approximating node weighted Steiner trees [10], can be improved to ln k... |

100 | A Nearly Best-possible Approximation Algorithm for Node-weighted Steiner Trees
- Klein, Ravi
- 1995
(Show Context)
Citation Context ...DT IME[n O(loglogn) ] [13, 6]. We give a 3 ln n approximation for the version when the vertices have weights. We also show that the upper bound of 2 ln k for approximating node weighted Steiner trees =-=[10]-=-, can be improved to ln k, when all Steiner vertices have unit weight. We then use this result to give a 3 ln k approximation for finding a connected dominating set for a specified subset of vertices.... |

100 | A tight analysis of the greedy algorithm for set cover - Slavı́k - 1997 |

81 | Bicriteria network design problems
- Marathe, Ravi, et al.
- 1995
(Show Context)
Citation Context ...aphs (defined in Subsection 1.2) for which no non-trivial approximation algorithm is known. The same is true for the CDS problem when the edges have weights. Recent work by Marathe, Ravi and Sundaram =-=[18] gives bic-=-riteria approximation algorithms for the "service-constrained network design problem". The goal is to design a tree that has low cost and is sufficiently close to each vertex in the graph (t... |

63 | Approximation algorithms for geometric tour and network design problems
- Mata, Mitchell
- 1995
(Show Context)
Citation Context ...strate that other generalizations of the CDS problem may be as hard to approximate as the "set TSP" problem for which no approximation algorithms are known. (For the Euclidean case, Mata and=-= Mitchell [14]-=- have given approximation algorithms for this problem.) c j V j Figure 3: Reduction of set TSP problem to edge weighted CDS Theorem 5.1 A polynomial approximation algorithm for the edge weighted conne... |

37 | Spanning trees with many leaves - Kleitman, West - 1991 |

37 | New approximation algorithms for the Steiner tree problem
- Karpinsky, Zelikovsky
- 1997
(Show Context)
Citation Context .... We also outline a second algorithm that gives an approximation factor of (1 + c)H(min(\Delta; k)) + O(1), where c is the best approximation ratio for the Steiner 2 tree problem (currently c = 1:644 =-=[12]-=-). Even though this algorithm has a better approximation guarantee, it is not practical due to the high running time, albeit polynomial. 1.2 Preliminaries The Steiner tree problem is defined as follow... |

23 |
Improved approximation algorithms for the Steiner tree problem
- Berman, Ramaiyer
- 1994
(Show Context)
Citation Context ...rms of the worst case approximation guarantee. However the first algorithms is simpler and faster. Most of the approximation algorithms that reduce the Steiner ratio below 2, have a high running time =-=[4, 12]-=-. 5 Lower Bounds 5.1 Hardness result for Connected Dominating Set We can prove that the set-cover problem can be reduced to the connected dominating set problem by an approximation preserving reductio... |

21 | R.: The power of local optimization: Approximation algorithms for maximum-leaf spanning tree
- Lu, Ravi
- 1992
(Show Context)
Citation Context ...emaining nodes of the graph form a spanning tree with many leaves. The maximum leaf spanning tree problem is another related problem that has been studied and factor 3 approximations are known for it =-=[16]-=-. Recently, Harary and Raghavachari [11] have shown that the email gossip number of a graph is exactly n \Gamma 1+OPTCDS , where OPTCDS is the size of the optimal connected dominating set. This indica... |

19 | Approximating the tree and tour covers of a graph
- Arkin, Halldorsson, et al.
- 1993
(Show Context)
Citation Context ...e graph (connected vertex cover). This is needed when one requires testing the links as well as the nodes. Approximation algorithms for the latter problem were given by Arkin, Halld'orsson and Hassin =-=[1]-=-. We observe that there is a simple algorithm for the unweighted connected vertex cover problem that gives a factor 2 approximation (the one given in [1] is more complicated). Do a Depth First Search,... |

16 |
A threshold of ln n for approximating set-cover
- Feige
- 1996
(Show Context)
Citation Context ...approximation preserving reduction from the set-cover problem to the connected dominating set problem, showing that it is hard to improve the approximation guarantee unless NP ` DT IME[n O(loglogn) ] =-=[13, 6]-=-. We give a 3 ln n approximation for the version when the vertices have weights. We also show that the upper bound of 2 ln k for approximating node weighted Steiner trees [10], can be improved to ln k... |

12 |
Depth first search and the vertex cover problem
- Savage
- 1982
(Show Context)
Citation Context ...omplicated). Do a Depth First Search, and take all the non-leaf vertices as the nodes in the vertex cover. This clearly induces a connected graph, and the approximation ratio is 2, as shown by Savage =-=[16]-=-. In practice, however this method will probably give large connected vertex covers. Other applications for the connected dominating set problem are in doing broadcasts for wireless computers in digit... |

12 |
Steiner trees, connected domination and strongly chordal graphs. Networks 15
- White, Farber, et al.
- 1985
(Show Context)
Citation Context ...cted dominating set problem has many interesting applications. Polynomial algorithms for the connected dominating set problem for special classes of graphs were given by White, Farber and Pulleyblank =-=[23]. 2 Algori-=-thm I We introduce an algorithm that finds a connected dominating set, by "growing" a tree. The idea behind the algorithm is the following: grow a tree T , starting from the vertex of maximu... |

6 |
Locating faults in a systematic manner in a large heterogeneous network
- Paul, Miller
- 1995
(Show Context)
Citation Context ...egree of a vertex in the graph. We use n and m to denote the number of vertices and edges in G. We use N(v) to denote the set of neighbors of a vertex v. 1.3 Applications The paper by Paul and Miller =-=[15] discusses-=- applications related to testing nodes in a computer network using a short "traveling tourist tour". They also consider the related question of finding a tour that visits each edge of the gr... |

6 | Computers and Intractability: A Guide tothe Theory of NP-Completeness - Garey, Johnson - 1979 |

5 |
personal communication
- Berman, Furer
- 1996
(Show Context)
Citation Context ...simple greedy strategy to connect the vertices in the dominating set, we proved a bound of H (\Delta) +H(H (\Delta)) [8]. Here we present a modification of the above algorithm, as suggested by Berman =-=[2]-=-, and are able to prove a performance guarantee of ln \Delta + 3. (Berman has an alternate proof for a performance ratio of H (\Delta) + 2.) 7 The algorithm runs in two phases. At the start of the fir... |

4 |
Improved approximation algorithms for node weighted Steiner trees, Manuscript
- Guha, Khuller
- 1997
(Show Context)
Citation Context ... n approximation for the node weighted CDS problem, where c n ln k is the approximation factor for the node weighted Steiner tree problem [13] and k is the number of terminals (currently c n = 1:6103 =-=[10]-=-). We next consider the Steiner CDS problem, where only a specified subset of vertices have to be dominated by a connected dominating set. For the unweighted case, we provide an approximation algorith... |

4 | Service-constrained network design problems - Marathe, Ravi, et al. - 1996 |

4 |
ik \A tight analysis of the greedy algorithm for set cover
- Slav
- 1996
(Show Context)
Citation Context ...g set in thesrst phase, and in the second phase connects the dominating set. In an earlier version of this paper [8] we established a bound of H() + H(H()). Using Slav ik's greedy set-cover bound =-=[17]-=-, we were able to show that the approximation factor is lnn+O(1). Recently, Berman suggested a modication to the algorithm, which improves the approximation factor to H() + 2. We describe this algor... |

3 |
Algorithms for unicast and multicast routing in ad-hoc networks", manuscript
- Kothari, Bharghavan
(Show Context)
Citation Context ...ces in the connected dominating set. The nodes in the connected dominating set are responsible for relaying messages. Each node not in the dominating set, is not responsible for relaying any messages =-=[9]-=-. Other relevant issues are regarding the maintenance of the connected dominating set as the network topology changes. 2 Algorithm I We introduce an algorithm that finds a connected dominating set, by... |

2 |
Ad-hoc routing using minimum connected dominating sets
- Bharghavan, Das
- 1997
(Show Context)
Citation Context ...t responsible for relaying any messages [12]. Many of the ideas in our paper have been used to design a distributed algorithm for routing based on minimum connected dominating sets in ad-hoc networks =-=[5]. Recent w-=-ork by Marathe, Ravi and Sundaram [18] gives bicriteria approximation algorithms for the "service-constrained network design problem". The goal is to design a tree that has low cost and is s... |

2 |
The E-mail gossip number and the connected domination number", manuscript
- Harary, Raghavachari
- 1996
(Show Context)
Citation Context ...ing tree with many leaves. The maximum leaf spanning tree problem is another related problem that has been studied and factor 3 approximations are known for it [16]. Recently, Harary and Raghavachari =-=[11]-=- have shown that the email gossip number of a graph is exactly n \Gamma 1+OPTCDS , where OPTCDS is the size of the optimal connected dominating set. This indicates that the connected dominating set pr... |

1 |
ik "A tight analysis of the greedy algorithm for set cover
- Slav
- 1996
(Show Context)
Citation Context ...irst phase, and in the second phase connects the dominating set. In an earlier version of this paper [8] we established a bound of H (\Delta) + H(H (\Delta)). Using Slav ' ik's greedy set-cover bound =-=[17]-=-, we were able to show that the approximation factor is ln n+O(1). Recently, Berman suggested a modification to the algorithm, which improves the approximation factor to H (\Delta) + 2. We describe th... |

1 | andM.Yannakakis, \On the hardnessofapproximating minimization problems - Lund - 1994 |