## Abstract

We consider the classical problem of scheduling parallel unrelated machines. Each job is to be processed by exactly one machine. Processing job j on machine i requires time P_{ij}. The goal is to find a schedule that minimizes the ℓ_{p} norm. Previous work showed a 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 a 2-approximation algorithm for any fixed ℓ_{p} norm (p > 1). This algorithm uses convex programming relaxation. We also give a √2-approximation algorithm for the ℓ _{2} norm. This algorithm relies on convex quadratic programming relaxation. To the best of our knowledge, this is the first time that general convex programming techniques (apart from SDPs and CQPs) are used in the area of scheduling. We show for any given ℓ_{p} norm a PTAS for any fixed number of machines. We also consider the multidimensional generalization of the problem in which the jobs are d-dimensional. Here the goal is to minimize the ℓ_{p} norm of the generalized load vector, which is a matrix where the rows represent the machines and the columns represent the jobs dimension. For this problem we give a (d+l)-approximation algorithm for any fixed ℓ_{p} norm (P > 1).

Original language | English |
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Pages (from-to) | 331-337 |

Number of pages | 7 |

Journal | Proceedings of the Annual ACM Symposium on Theory of Computing |

DOIs | |

State | Published - 2005 |

Event | 13th Color Imaging Conference: Color Science, Systems, Technologies, and Applications - Scottsdale, AZ, United States Duration: 7 Nov 2005 → 11 Nov 2005 |

## Keywords

- Approximation algorithms
- Convex programming
- Randomized algorithms
- Scheduling unrelated parallel machines