### 抜粋

Consider the longest path problem for directed acyclic graphs (DAGs), where a mutually independent random variable is associated with each of the edges as its edge length. Given a DAG G and any distributions that the random variables obey, let F _{MAX}(x) be the distribution function of the longest path length.We first represent F _{MAX}(x) by a repeated integral that involves n-1 integrals, where n is the order of G. We next present an algorithm to symbolically execute the repeated integral, provided that the random variables obey the standard exponential distribution. Although there can be ω(2 ^{n}) paths in G, its running time is bounded by a polynomial in n, provided that k, the cardinality of the maximum anti-chain of the incidence graph of G, is bounded by a constant. We finally propose an algorithm that takes x and e > 0 as inputs and approximates the value of repeated integral of x, assuming that the edge length distributions satisfy the following three natural conditions: (1) The length of each edge (vi, vj) e E is non-negative, (2) the Taylor series of its distribution function Fij (x) converges to Fij (x), and (3) there is a constant σ that satisfies σ ^{p} ≤ (d/dx) ^{p} Fij(x) for any non-negative integer p. It runs in polynomial time in n, and its error is bounded by e, when x, e, σ and k can be regarded as constants.

元の言語 | 英語 |
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ホスト出版物のタイトル | Theory and Applications of Models of Computation - 6th Annual Conference, TAMC 2009, Proceedings |

ページ | 98-107 |

ページ数 | 10 |

DOI | |

出版物ステータス | 出版済み - 7 16 2009 |

イベント | 6th Annual Conference on Theory and Applications of Models of Computation, TAMC 2009 - Changsha, 中国 継続期間: 5 18 2009 → 5 22 2009 |

### 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) |
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巻 | 5532 LNCS |

ISSN（印刷物） | 0302-9743 |

ISSN（電子版） | 1611-3349 |

### その他

その他 | 6th Annual Conference on Theory and Applications of Models of Computation, TAMC 2009 |
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国 | 中国 |

市 | Changsha |

期間 | 5/18/09 → 5/22/09 |

### フィンガープリント

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### これを引用

*Theory and Applications of Models of Computation - 6th Annual Conference, TAMC 2009, Proceedings*(pp. 98-107). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); 巻数 5532 LNCS). https://doi.org/10.1007/978-3-642-02017-9_13