## Abstract

Motivated by fundamental optimization problems in video delivery over wireless networks, we consider the following problem of packing resizable items (PRI). Given is a bin of capacity B>0, and a set I of items. Each item j∈I has a size s_{j}>0. A packed item must stay in the bin for a fixed common time interval. To accommodate additional items in the bin, each item j can be compressed to a given size p_{j}∈[0, s_{j}) for at most a fraction q_{j}∈[0, 1) of the packing interval, during one continuous sub-interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI is a generalization of the well-known Knapsack problem, and it is strongly NP-hard already for highly restricted instances.Our main result is an approximation algorithm that packs, for any instance I of PRI, at least 23OPT(I)-2 items, where OPT(I) is the number of items packed in an optimal solution. Our algorithm yields a better performance ratio for instances in which the maximum compression time of an item is qmax∈(0,12). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OPT(I). - 2 items. Finally, we show that a non-trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS).

Original language | English |
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Pages (from-to) | 91-105 |

Number of pages | 15 |

Journal | Theoretical Computer Science |

Volume | 553 |

Issue number | C |

DOIs | |

State | Published - 2014 |

### Bibliographical note

Publisher Copyright:© 2013 Elsevier B.V.

## Keywords

- Approximation algorithms
- Packing
- Variable quality-of-service

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)