### Abstract

Given a (directed or undirected) graph G = (V,E), a mutually independent random variable X_{e} obeying a normal distribution for each edge e ∈ E that represents its edge weight, and a property P on graph, a stochastic graph maximization problem asks the distribution function F _{MAX}(x) of random variable X_{MAX} = max_{P∈P} Σ_{e∈A} X_{e}, where property P is identified with the set of subgraphs P = (U,A) of G having P. This paper proposes a generic algorithm for computing an elementary function F̃(x) that approximates F_{MAX}(x). It is applicable to any P and runs in time O(T _{AMAX}(P)+T_{ACNT}(P)), provided the existence of an algorithm A_{MAX} that solves the (deterministic) graph maximization problem for P in time T_{AMAX}(P) and an algorithm A_{CNT} that outputs an upper bound on |P| in time T_{ACNT}(P).We analyze the approximation ratio and apply it to three graph maximization problems. In case no efficient algorithms are known for solving the graph maximization problem for P, an approximation algorithm A_{APR} can be used instead of A_{MAX} to reduce the time complexity, at the expense of increase of approximation ratio. Our algorithm can be modified to handle minimization problems.

Original language | English |
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Title of host publication | Stochastic Algorithms |

Subtitle of host publication | Foundations and Applications - 5th International Symposium, SAGA 2009, Proceedings |

Pages | 89-103 |

Number of pages | 15 |

DOIs | |

Publication status | Published - Dec 1 2009 |

Event | 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009 - Sapporo, Japan Duration: Oct 26 2009 → Oct 28 2009 |

### 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 | 5792 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009 |
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Country | Japan |

City | Sapporo |

Period | 10/26/09 → 10/28/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Stochastic Algorithms: Foundations and Applications - 5th International Symposium, SAGA 2009, Proceedings*(pp. 89-103). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5792 LNCS). https://doi.org/10.1007/978-3-642-04944-6_8

**A generic algorithm for approximately solving stochastic graph optimization problems.** / Ando, Ei; Ono, Hirotaka; Yamashita, Masafumi.

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

*Stochastic Algorithms: Foundations and Applications - 5th International Symposium, SAGA 2009, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5792 LNCS, pp. 89-103, 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009, Sapporo, Japan, 10/26/09. https://doi.org/10.1007/978-3-642-04944-6_8

}

TY - GEN

T1 - A generic algorithm for approximately solving stochastic graph optimization problems

AU - Ando, Ei

AU - Ono, Hirotaka

AU - Yamashita, Masafumi

PY - 2009/12/1

Y1 - 2009/12/1

N2 - Given a (directed or undirected) graph G = (V,E), a mutually independent random variable Xe obeying a normal distribution for each edge e ∈ E that represents its edge weight, and a property P on graph, a stochastic graph maximization problem asks the distribution function F MAX(x) of random variable XMAX = maxP∈P Σe∈A Xe, where property P is identified with the set of subgraphs P = (U,A) of G having P. This paper proposes a generic algorithm for computing an elementary function F̃(x) that approximates FMAX(x). It is applicable to any P and runs in time O(T AMAX(P)+TACNT(P)), provided the existence of an algorithm AMAX that solves the (deterministic) graph maximization problem for P in time TAMAX(P) and an algorithm ACNT that outputs an upper bound on |P| in time TACNT(P).We analyze the approximation ratio and apply it to three graph maximization problems. In case no efficient algorithms are known for solving the graph maximization problem for P, an approximation algorithm AAPR can be used instead of AMAX to reduce the time complexity, at the expense of increase of approximation ratio. Our algorithm can be modified to handle minimization problems.

AB - Given a (directed or undirected) graph G = (V,E), a mutually independent random variable Xe obeying a normal distribution for each edge e ∈ E that represents its edge weight, and a property P on graph, a stochastic graph maximization problem asks the distribution function F MAX(x) of random variable XMAX = maxP∈P Σe∈A Xe, where property P is identified with the set of subgraphs P = (U,A) of G having P. This paper proposes a generic algorithm for computing an elementary function F̃(x) that approximates FMAX(x). It is applicable to any P and runs in time O(T AMAX(P)+TACNT(P)), provided the existence of an algorithm AMAX that solves the (deterministic) graph maximization problem for P in time TAMAX(P) and an algorithm ACNT that outputs an upper bound on |P| in time TACNT(P).We analyze the approximation ratio and apply it to three graph maximization problems. In case no efficient algorithms are known for solving the graph maximization problem for P, an approximation algorithm AAPR can be used instead of AMAX to reduce the time complexity, at the expense of increase of approximation ratio. Our algorithm can be modified to handle minimization problems.

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U2 - 10.1007/978-3-642-04944-6_8

DO - 10.1007/978-3-642-04944-6_8

M3 - Conference contribution

AN - SCOPUS:78650651188

SN - 3642049435

SN - 9783642049439

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

SP - 89

EP - 103

BT - Stochastic Algorithms

ER -