### 抄録

Given an n-vertex graph G = (V,E) and a set R ⊆ {{x,y}|x,y ∈ V} of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex, which is originally motivated by the design of secure communications in distributed computing. This problem has been shown to be LOGAPX-hard for general directed topologies of G and R. In this paper, we consider the complexity of MCD for more practical topologies of G and R, that is, when G or R forms an (undirected) tree; tree structures are frequently adopted to construct efficient communication networks. We first show that MCD is still APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree Δ of the tree: MCD for tree request set with constant Δ is solvable in polynomial time, while that with Δ = Ω(n) is 2.78-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that if G is a tree, the problem can be solved in O(n ^{1+ε}|R|), where ε is an arbitrarily small positive constant number.

元の言語 | 英語 |
---|---|

ホスト出版物のタイトル | Theory and Applications of Models of Computation - 9th Annual Conference, TAMC 2012, Proceedings |

ページ | 548-559 |

ページ数 | 12 |

DOI | |

出版物ステータス | 出版済み - 5 18 2012 |

イベント | 9th Annual Conference on Theory and Applications of Models of Computation, TAMC 2012 - Beijing, 中国 継続期間: 5 16 2012 → 5 21 2012 |

### 出版物シリーズ

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

巻 | 7287 LNCS |

ISSN（印刷物） | 0302-9743 |

ISSN（電子版） | 1611-3349 |

### その他

その他 | 9th Annual Conference on Theory and Applications of Models of Computation, TAMC 2012 |
---|---|

国 | 中国 |

市 | Beijing |

期間 | 5/16/12 → 5/21/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### これを引用

*Theory and Applications of Models of Computation - 9th Annual Conference, TAMC 2012, Proceedings*(pp. 548-559). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); 巻数 7287 LNCS). https://doi.org/10.1007/978-3-642-29952-0_51

**Minimum certificate dispersal with tree structures.** / Izumi, Taisuke; Izumi, Tomoko; Ono, Hirotaka; Wada, Koichi.

研究成果: 著書/レポートタイプへの貢献 › 会議での発言

*Theory and Applications of Models of Computation - 9th Annual Conference, TAMC 2012, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 巻. 7287 LNCS, pp. 548-559, 9th Annual Conference on Theory and Applications of Models of Computation, TAMC 2012, Beijing, 中国, 5/16/12. https://doi.org/10.1007/978-3-642-29952-0_51

}

TY - GEN

T1 - Minimum certificate dispersal with tree structures

AU - Izumi, Taisuke

AU - Izumi, Tomoko

AU - Ono, Hirotaka

AU - Wada, Koichi

PY - 2012/5/18

Y1 - 2012/5/18

N2 - Given an n-vertex graph G = (V,E) and a set R ⊆ {{x,y}|x,y ∈ V} of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex, which is originally motivated by the design of secure communications in distributed computing. This problem has been shown to be LOGAPX-hard for general directed topologies of G and R. In this paper, we consider the complexity of MCD for more practical topologies of G and R, that is, when G or R forms an (undirected) tree; tree structures are frequently adopted to construct efficient communication networks. We first show that MCD is still APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree Δ of the tree: MCD for tree request set with constant Δ is solvable in polynomial time, while that with Δ = Ω(n) is 2.78-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that if G is a tree, the problem can be solved in O(n 1+ε|R|), where ε is an arbitrarily small positive constant number.

AB - Given an n-vertex graph G = (V,E) and a set R ⊆ {{x,y}|x,y ∈ V} of requests, we consider to assign a set of edges to each vertex in G so that for every request {u, v} in R the union of the edge sets assigned to u and v contains a path from u to v. The Minimum Certificate Dispersal Problem (MCD) is defined as one to find an assignment that minimizes the sum of the cardinality of the edge set assigned to each vertex, which is originally motivated by the design of secure communications in distributed computing. This problem has been shown to be LOGAPX-hard for general directed topologies of G and R. In this paper, we consider the complexity of MCD for more practical topologies of G and R, that is, when G or R forms an (undirected) tree; tree structures are frequently adopted to construct efficient communication networks. We first show that MCD is still APX-hard when R is a tree, even a star. We then explore the problem from the viewpoint of the maximum degree Δ of the tree: MCD for tree request set with constant Δ is solvable in polynomial time, while that with Δ = Ω(n) is 2.78-approximable in polynomial time but hard to approximate within 1.01 unless P=NP. As for the structure of G itself, we show that if G is a tree, the problem can be solved in O(n 1+ε|R|), where ε is an arbitrarily small positive constant number.

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U2 - 10.1007/978-3-642-29952-0_51

DO - 10.1007/978-3-642-29952-0_51

M3 - Conference contribution

AN - SCOPUS:84861010425

SN - 9783642299513

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

SP - 548

EP - 559

BT - Theory and Applications of Models of Computation - 9th Annual Conference, TAMC 2012, Proceedings

ER -