### 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) |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Management Information Systems
- Strategy and Management
- Management Science and Operations Research
- Marketing

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## Cite this

*Journal of the Operational Research Society*,

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