## Abstract

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ_{p} norms. We address this problem by introducing the concept of an All-norm ρ-approximation algorithm, which supplies one solution that guarantees ρ-approximation to all ℓ_{p} norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each is associatedwith a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al. [12] showeda 2-approximation algorithm for the problem with respect to the ℓ_{∞} norm. For any fixed ℓ_{p} norm the previously known approximation algorithm has a performance of θ(p). We provide an all-norm 2-approximation polynomial algorithm for the restrictedassignment problem. On the other hand, we show that for any given ℓ_{p} norm (p > 1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given ℓ_{p} norm a FPTAS for any fixedn umber of machines.

Original language | English |
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Title of host publication | Algorithm Theory - SWAT 2002 - 8th Scandinavian Workshop on Algorithm Theory, Proceedings |

Editors | Martti Penttonen, Erik Meineche Schmidt |

Publisher | Springer Verlag |

Pages | 288-297 |

Number of pages | 10 |

ISBN (Print) | 9783540438663 |

DOIs | |

State | Published - 2002 |

Externally published | Yes |

Event | 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002 - Turku, Finland Duration: 3 Jul 2002 → 5 Jul 2002 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2368 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002 |
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Country/Territory | Finland |

City | Turku |

Period | 3/07/02 → 5/07/02 |

### Bibliographical note

Funding Information:* Corresponding author. E-mail addresses: azar@tau.ac.il (Y. Azar), lea@idc.ac.il (L. Epstein), yo@tau.ac.il (Y. Richter), g.j.woeginger@math.utwente.nl (G.J. Woeginger). 1 Research supported in part by the Israeli Ministry of industry and trade and by the Israel Science Foundation. 2 Research supported in part by the Israel Science Foundation. 3 Research supported in part by the Israeli Ministry of industry and trade. 4 Supported by the START program Y43-MAT of the Austrian Ministry of Science.

Publisher Copyright:

© Springer-Verlag Berlin Heidelberg 2002.

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)