### Abstract

Let P = [p1, p2,...., p_{n}] be a set of n terminal points in the Euclidean plane, where point p_{i} has a service request of grade g(p_{i}) ∈ (1, 2,..., n). Let 0 < c(1) < c(2) < ⋯ < c(n) be n real numbers. The Grade of Service Steiner Minimum Tree (GOSST) problem asks for a minimum cost network interconnecting point set P and some Steiner points with a service request of grade 0 such that (1) between each pair of terminal points p_{i} and p_{j} there is a path whose minimum grade of service is at least as large as min(g(p_{i}), g(p_{j})); and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g). The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio 4/3ρ (((5+4√2)/7)ρ, respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1 + ε-approximation to the minimum cost network under a given topology or its degeneracies in O(n^{1.5}(log n+log(1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k-optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.

Original language | English (US) |
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Pages (from-to) | 479-500 |

Number of pages | 22 |

Journal | Algorithmica (New York) |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2001 |

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### Keywords

- Approximation algorithms
- Communication networks
- Grade of service Steiner minimum trees
- Interior-point algorithms
- K-Optimal heuristic
- Transportation networks

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica (New York)*,

*31*(4), 479-500. https://doi.org/10.1007/s00453-001-0050-6