### Abstract

Given a graph G = (V,E) and a set of vertices M ⊑ V, a vertex v ϵ V is said to be controlled by M if the majority of v's neighbors (including itself) belongs to M. M is called a monopoly if every vertex v ϵ V is controlled by M. For a specified M and a range for E (E_{1} ⊑ E ⊑ E_{2}), we try to determine E such that M is a monopoly in G = (V,E). We first present a polynomial algorithm for testing if such an E exists, by formulating it as a network flow problem. Assuming that a solution E does exist, we then show that a solution with the maximum or minimum |E| can be found in polynomial time, by considering them as weighted matching problems. In case there is no solution E, we want to maximize the number of vertices controlled by the given M. Unfortunately, this problem turns out to be NP-hard. We therefore design a simple approximation algorithm which guarantees an approximation ratio of 2.

Original language | English |
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Title of host publication | Algorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings |

Editors | Magnús M. Halldórsson |

Publisher | Springer Verlag |

Pages | 513-526 |

Number of pages | 14 |

ISBN (Print) | 3540676902, 9783540676904 |

DOIs | |

Publication status | Published - Jan 1 2000 |

Event | 7th Scandinavian Workshop on Algorithm Theory, SWAT 2000 - Bergen, Norway Duration: Jul 5 2000 → Jul 7 2000 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1851 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 7th Scandinavian Workshop on Algorithm Theory, SWAT 2000 |
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Country | Norway |

City | Bergen |

Period | 7/5/00 → 7/7/00 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings*(pp. 513-526). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1851). Springer Verlag. https://doi.org/10.1007/3-540-44985-X

**Max-and min-neighborhood monopolies.** / Makino, Kazuhisa; Yamashita, Masafumi; Kameda, Tiko.

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

*Algorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1851, Springer Verlag, pp. 513-526, 7th Scandinavian Workshop on Algorithm Theory, SWAT 2000, Bergen, Norway, 7/5/00. https://doi.org/10.1007/3-540-44985-X

}

TY - GEN

T1 - Max-and min-neighborhood monopolies

AU - Makino, Kazuhisa

AU - Yamashita, Masafumi

AU - Kameda, Tiko

PY - 2000/1/1

Y1 - 2000/1/1

N2 - Given a graph G = (V,E) and a set of vertices M ⊑ V, a vertex v ϵ V is said to be controlled by M if the majority of v's neighbors (including itself) belongs to M. M is called a monopoly if every vertex v ϵ V is controlled by M. For a specified M and a range for E (E1 ⊑ E ⊑ E2), we try to determine E such that M is a monopoly in G = (V,E). We first present a polynomial algorithm for testing if such an E exists, by formulating it as a network flow problem. Assuming that a solution E does exist, we then show that a solution with the maximum or minimum |E| can be found in polynomial time, by considering them as weighted matching problems. In case there is no solution E, we want to maximize the number of vertices controlled by the given M. Unfortunately, this problem turns out to be NP-hard. We therefore design a simple approximation algorithm which guarantees an approximation ratio of 2.

AB - Given a graph G = (V,E) and a set of vertices M ⊑ V, a vertex v ϵ V is said to be controlled by M if the majority of v's neighbors (including itself) belongs to M. M is called a monopoly if every vertex v ϵ V is controlled by M. For a specified M and a range for E (E1 ⊑ E ⊑ E2), we try to determine E such that M is a monopoly in G = (V,E). We first present a polynomial algorithm for testing if such an E exists, by formulating it as a network flow problem. Assuming that a solution E does exist, we then show that a solution with the maximum or minimum |E| can be found in polynomial time, by considering them as weighted matching problems. In case there is no solution E, we want to maximize the number of vertices controlled by the given M. Unfortunately, this problem turns out to be NP-hard. We therefore design a simple approximation algorithm which guarantees an approximation ratio of 2.

UR - http://www.scopus.com/inward/record.url?scp=84956869114&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956869114&partnerID=8YFLogxK

U2 - 10.1007/3-540-44985-X

DO - 10.1007/3-540-44985-X

M3 - Conference contribution

AN - SCOPUS:84956869114

SN - 3540676902

SN - 9783540676904

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 513

EP - 526

BT - Algorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings

A2 - Halldórsson, Magnús M.

PB - Springer Verlag

ER -