## 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) |
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Pages (from-to) | 22-42 |

Number of pages | 21 |

Journal | SIAM Journal on Optimization |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - 1999 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Applied Mathematics