### Abstract

We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.

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

Pages (from-to) | 382-389 |

Number of pages | 8 |

Journal | Journal of the Operational Research Society |

Volume | 56 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2005 |

### Fingerprint

### Keywords

- Branch-and-bound: Lagrangian relaxation
- Minimum spanning tree

### ASJC Scopus subject areas

- Management of Technology and Innovation
- Strategy and Management
- Management Science and Operations Research

### Cite this

*Journal of the Operational Research Society*,

*56*(4), 382-389. https://doi.org/10.1057/palgrave.jors.2601821

**Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm.** / Haouari, M.; Chaouachi, J.; Dror, Moshe.

Research output: Contribution to journal › Article

*Journal of the Operational Research Society*, vol. 56, no. 4, pp. 382-389. https://doi.org/10.1057/palgrave.jors.2601821

}

TY - JOUR

T1 - Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm

AU - Haouari, M.

AU - Chaouachi, J.

AU - Dror, Moshe

PY - 2005/4

Y1 - 2005/4

N2 - We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.

AB - We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.

KW - Branch-and-bound: Lagrangian relaxation

KW - Minimum spanning tree

UR - http://www.scopus.com/inward/record.url?scp=15844376710&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15844376710&partnerID=8YFLogxK

U2 - 10.1057/palgrave.jors.2601821

DO - 10.1057/palgrave.jors.2601821

M3 - Article

VL - 56

SP - 382

EP - 389

JO - Journal of the Operational Research Society

JF - Journal of the Operational Research Society

SN - 0160-5682

IS - 4

ER -