### Abstract

In this paper, we study the problem of computing the minimum cost pipe network interconnecting a given set of wells and a treatment site, where each well has a given capacity and the treatment site has a capacity that is no less than the sum of all the capacities of the wells. This is a generalized Steiner minimum tree problem which has applications in communication networks and in groundwater treatment. We prove that there exists a minimum cost pipe network that is the minimum cost network under a full Steiner topology. For each given full Steiner topology, we can compute all the edge weights in linear time. A powerful interior-point algorithm is then used to find the minimum cost network under this given topology. We also prove a lower bound theorem which enables pruning in a backtrack method that partially enumerates the full Steiner topologies in search for a minimum cost pipe network. A heuristic ordering algorithm is proposed to enhance the performance of the backtrack algorithm. We then define the notion of k-optimality and present an efficient (polynomial time) algorithm for checking 5-optimality. We present a 5-optimal heuristic algorithm for computing good solutions when the problem size is too large for the exact algorithm. Computational results are presented.

Original language | English (US) |
---|---|

Pages (from-to) | 22-42 |

Number of pages | 21 |

Journal | SIAM Journal on Optimization |

Volume | 10 |

Issue number | 1 |

State | Published - 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

- Backtrack
- Bounding theorem
- Generalized steiner minimum tree problem
- Interior-point methods
- k-optimal
- Minimum cost pipe network

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Optimization*,

*10*(1), 22-42.

**Computing the minimum cost pipe network interconnecting one sink and many sources.** / Xue, Guoliang; Lillys, Theodore P.; Dougherty, David E.

Research output: Contribution to journal › Article

*SIAM Journal on Optimization*, vol. 10, no. 1, pp. 22-42.

}

TY - JOUR

T1 - Computing the minimum cost pipe network interconnecting one sink and many sources

AU - Xue, Guoliang

AU - Lillys, Theodore P.

AU - Dougherty, David E.

PY - 1999

Y1 - 1999

N2 - In this paper, we study the problem of computing the minimum cost pipe network interconnecting a given set of wells and a treatment site, where each well has a given capacity and the treatment site has a capacity that is no less than the sum of all the capacities of the wells. This is a generalized Steiner minimum tree problem which has applications in communication networks and in groundwater treatment. We prove that there exists a minimum cost pipe network that is the minimum cost network under a full Steiner topology. For each given full Steiner topology, we can compute all the edge weights in linear time. A powerful interior-point algorithm is then used to find the minimum cost network under this given topology. We also prove a lower bound theorem which enables pruning in a backtrack method that partially enumerates the full Steiner topologies in search for a minimum cost pipe network. A heuristic ordering algorithm is proposed to enhance the performance of the backtrack algorithm. We then define the notion of k-optimality and present an efficient (polynomial time) algorithm for checking 5-optimality. We present a 5-optimal heuristic algorithm for computing good solutions when the problem size is too large for the exact algorithm. Computational results are presented.

AB - In this paper, we study the problem of computing the minimum cost pipe network interconnecting a given set of wells and a treatment site, where each well has a given capacity and the treatment site has a capacity that is no less than the sum of all the capacities of the wells. This is a generalized Steiner minimum tree problem which has applications in communication networks and in groundwater treatment. We prove that there exists a minimum cost pipe network that is the minimum cost network under a full Steiner topology. For each given full Steiner topology, we can compute all the edge weights in linear time. A powerful interior-point algorithm is then used to find the minimum cost network under this given topology. We also prove a lower bound theorem which enables pruning in a backtrack method that partially enumerates the full Steiner topologies in search for a minimum cost pipe network. A heuristic ordering algorithm is proposed to enhance the performance of the backtrack algorithm. We then define the notion of k-optimality and present an efficient (polynomial time) algorithm for checking 5-optimality. We present a 5-optimal heuristic algorithm for computing good solutions when the problem size is too large for the exact algorithm. Computational results are presented.

KW - Backtrack

KW - Bounding theorem

KW - Generalized steiner minimum tree problem

KW - Interior-point methods

KW - k-optimal

KW - Minimum cost pipe network

UR - http://www.scopus.com/inward/record.url?scp=0033261255&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033261255&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033261255

VL - 10

SP - 22

EP - 42

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 1

ER -