### Abstract

We consider new variants of the vertex/edge domination problems on graphs. A vertex is said to dominate itself and its all adjacent vertices, and similarly an edge is said to dominate itself and its all adjacent edges. Given an input graph G = (V;E) and an integer k, the κ-Vertex (κ-Edge) Maximum Domination (κ-MaxVD and κ-MaxED, respectively) is to find a subset D _{V} ⊆ V of vertices (resp., D _{E} ⊆ E of edges) with size at most k that maximizes the cardinality of dominated vertices (resp., edges). In this paper, we first show that a simple greedy strategy achieves an approximation ratio of (1 - 1=e) for both k-MaxVD and κ-MaxED. Then, we show that this approximation ratio is the best possible for κ-MaxVD unless P = NP. We also prove that, for any constant ε > 0, there is no polynomial time 1303=1304+ε approximation algorithm for κ-MaxED unless P = NP. However, if k is not larger than the size of the minimum maximal matching, κ-MaxED is 3/4-approximable in polynomial time.

Original language | English |
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Title of host publication | Theory of Computing 2011 - Proceedings of the 17th Computing |

Subtitle of host publication | The Australasian Theory Symposium, CATS 2011 |

Pages | 55-61 |

Number of pages | 7 |

Publication status | Published - Dec 1 2011 |

Event | Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011 - Perth, WA, Australia Duration: Jan 17 2011 → Jan 20 2011 |

### Publication series

Name | Conferences in Research and Practice in Information Technology Series |
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Volume | 119 |

ISSN (Print) | 1445-1336 |

### Other

Other | Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011 |
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Country | Australia |

City | Perth, WA |

Period | 1/17/11 → 1/20/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Computer Science Applications
- Hardware and Architecture
- Information Systems
- Software

### Cite this

*Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011*(pp. 55-61). (Conferences in Research and Practice in Information Technology Series; Vol. 119).

**Maximum Domination problem.** / Miyano, Eiji; Ono, Hirotaka.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011.*Conferences in Research and Practice in Information Technology Series, vol. 119, pp. 55-61, Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011, Perth, WA, Australia, 1/17/11.

}

TY - GEN

T1 - Maximum Domination problem

AU - Miyano, Eiji

AU - Ono, Hirotaka

PY - 2011/12/1

Y1 - 2011/12/1

N2 - We consider new variants of the vertex/edge domination problems on graphs. A vertex is said to dominate itself and its all adjacent vertices, and similarly an edge is said to dominate itself and its all adjacent edges. Given an input graph G = (V;E) and an integer k, the κ-Vertex (κ-Edge) Maximum Domination (κ-MaxVD and κ-MaxED, respectively) is to find a subset D V ⊆ V of vertices (resp., D E ⊆ E of edges) with size at most k that maximizes the cardinality of dominated vertices (resp., edges). In this paper, we first show that a simple greedy strategy achieves an approximation ratio of (1 - 1=e) for both k-MaxVD and κ-MaxED. Then, we show that this approximation ratio is the best possible for κ-MaxVD unless P = NP. We also prove that, for any constant ε > 0, there is no polynomial time 1303=1304+ε approximation algorithm for κ-MaxED unless P = NP. However, if k is not larger than the size of the minimum maximal matching, κ-MaxED is 3/4-approximable in polynomial time.

AB - We consider new variants of the vertex/edge domination problems on graphs. A vertex is said to dominate itself and its all adjacent vertices, and similarly an edge is said to dominate itself and its all adjacent edges. Given an input graph G = (V;E) and an integer k, the κ-Vertex (κ-Edge) Maximum Domination (κ-MaxVD and κ-MaxED, respectively) is to find a subset D V ⊆ V of vertices (resp., D E ⊆ E of edges) with size at most k that maximizes the cardinality of dominated vertices (resp., edges). In this paper, we first show that a simple greedy strategy achieves an approximation ratio of (1 - 1=e) for both k-MaxVD and κ-MaxED. Then, we show that this approximation ratio is the best possible for κ-MaxVD unless P = NP. We also prove that, for any constant ε > 0, there is no polynomial time 1303=1304+ε approximation algorithm for κ-MaxED unless P = NP. However, if k is not larger than the size of the minimum maximal matching, κ-MaxED is 3/4-approximable in polynomial time.

UR - http://www.scopus.com/inward/record.url?scp=84864629153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864629153&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84864629153

SN - 9781920682989

T3 - Conferences in Research and Practice in Information Technology Series

SP - 55

EP - 61

BT - Theory of Computing 2011 - Proceedings of the 17th Computing

ER -