### Abstract

Buy-at-bulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buy-at-bulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first poly-logarithmic approximation ratio for the non-uniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log^{4} h), where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Poly-logarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems.

Original language | English (US) |
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Pages (from-to) | 1772-1798 |

Number of pages | 27 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 5 |

DOIs | |

State | Published - Feb 25 2010 |

### Keywords

- Approximation algorithm
- Concave cost
- Economies of scale
- Network design
- Network flow
- Nonuniform buy-at-bulk

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*39*(5), 1772-1798. https://doi.org/10.1137/090750317