## Abstract

We study the storage allocation problem (SAP) which is a variant of the unsplittable flow problem on paths (UFPP). A SAP instance consists of a path P= (V, E) and a set J of tasks. Each edge e∈ E has a capacity c_{e} and each task j∈ J is associated with a path I_{j} in P, a demand d_{j} and a weight w_{j}. The goal is to find a maximum weight subset S⊆ J of tasks and a height function h: S→ R^{+} such that (i) h(j) + d_{j}≤ c_{e}, for every e∈ I_{j}; and (ii) if j, i∈ S such that I_{j}∩ I_{i}≠ ∅ and h(j) ≥ h(i) , then h(j) ≥ h(i) + d_{i}. SAP can be seen as a rectangle packing problem in which rectangles can be moved vertically, but not horizontally. We present a polynomial time (9 + ε) -approximation algorithm for SAP. Our algorithm is based on a variation of the framework for approximating UFPP by Bonsma et al. [FOCS 2011] and on a (4 + ε) -approximation algorithm for δ-small SAP instances (in which d_{j}≤ δ· c_{e}, for every e∈ I_{j}, for a sufficiently small constant δ> 0). In our algorithm for δ-small instances, tasks are packed carefully in strips in a UFPP manner, and then a (1 + ε) factor is incurred by a reduction from SAP to UFPP in strips. The strips are stacked to form a SAP solution. Finally, we provide a (10 + ε) -approximation algorithm for SAP on ring networks.

Original language | English |
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Pages (from-to) | 1105-1127 |

Number of pages | 23 |

Journal | Algorithmica |

Volume | 77 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2017 |

### Bibliographical note

Publisher Copyright:© 2016, Springer Science+Business Media New York.

## Keywords

- Approximation algorithms
- Bandwidth allocation
- Rectangle packing
- Storage allocation
- Unsplittable flow