### Abstract

Consider a matroid where each element has a real-valued cost and a color, red or green; a base is sought that contains q red elements and has smallest possible cost. An algorithm for the problem on general matroids is presented, along with a number of variations. Its efficiency is demonstrated by implementations on specific matroids. In all cases but one, the running time matches the best-known algorithm for the problem without the red element constraint: On graphic matroids, a smallest spanning tree with q red edges can be found in time O(n log n) more than what is needed to find a minimum spanning tree. A special case is finding a smallest spanning tree with a degree constraint; here the time is only O(m + n) more than that needed to find one minimum spanning tree. On transversal and matching matroids, the time is the same as the best-known algorithms for a minimum cost base. This also holds for transversal matroids for convex graphs, which model a scheduling problem on unit-length jobs with release times and deadlines. On partition matroids, a linear-time algorithm is presented. Finally an algorithm related to our general approach finds a smallest spanning tree on a directed graph, where the given root has a degree constraint. Again the time matches the best-known algorithm for the problem without the red element (i.e., degree) constraint.

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

Pages (from-to) | 80-131 |

Number of pages | 52 |

Journal | Journal of Algorithms |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1984 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Computational Mathematics
- Computational Theory and Mathematics

## Fingerprint Dive into the research topics of 'Efficient algorithms for a family of matroid intersection problems'. Together they form a unique fingerprint.

## Cite this

*Journal of Algorithms*,

*5*(1), 80-131. https://doi.org/10.1016/0196-6774(84)90042-7