### Abstract

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.

Original language | English |
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Title of host publication | Theory and Applications of Models of Computation - 9th Annual Conference, TAMC 2012, Proceedings |

Pages | 548-559 |

Number of pages | 12 |

DOIs | |

Publication status | Published - May 18 2012 |

Event | 9th Annual Conference on Theory and Applications of Models of Computation, TAMC 2012 - Beijing, China Duration: May 16 2012 → May 21 2012 |

### 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 | 7287 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th Annual Conference on Theory and Applications of Models of Computation, TAMC 2012 |
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Country | China |

City | Beijing |

Period | 5/16/12 → 5/21/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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); Vol. 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.

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

*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), vol. 7287 LNCS, pp. 548-559, 9th Annual Conference on Theory and Applications of Models of Computation, TAMC 2012, Beijing, China, 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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UR - http://www.scopus.com/inward/citedby.url?scp=84861010425&partnerID=8YFLogxK

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

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

M3 - Conference contribution

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 -