## Abstract

The General Scheduling Problem (GSP) generalizes scheduling problems with sum of cost objectives such as weighted flow time and weighted tardiness. Given a set of jobs with processing times, release dates, and job dependent cost functions, we seek to find a minimum cost preemptive schedule on a single machine. The best known algorithm for this problem and also for weighted flow time/tardiness is an O(log log P)-approximation (where P denotes the range of the job processing times), while the best lower bound shows only strong NP-hardness. When release dates are identical there is also a gap: the problem remains strongly NP-hard and the best known approximation algorithm has a ratio of e+ϵ (running in quasi-polynomial time). We reduce the latter gap by giving a QPTAS if the numbers in the input are quasi-polynomially bounded, ruling out the existence of an APX-hardness proof unless NP ⊆ DTIME(2^{polylog(n)}). Our techniques are based on the QPTAS known for the UFP-Cover problem, a particular case of GSP where we must pick a subset of intervals (jobs) on the real line with associated heights and costs. If an interval is selected, its height will help cover a given demand on any point contained within the interval. We reduce our problem to a generalization of UFP-Cover and use a sophisticated divide-and-conquer procedure with interdependent non-symmetric subproblems. We also present a pseudo-polynomial time approximation scheme for two variants of UFPCover. For the case of agreeable intervals we give an algorithm based on a new dynamic programming approach which might be useful for other problems of this type. The second one is a resource augmentation setting where we are allowed to slightly enlarge each interval.

Original language | English |
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Title of host publication | 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 |

Editors | Anca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis |

Publisher | Dagstuhl |

Pages | 31:1-31:14 |

Number of pages | 14 |

ISBN (Electronic) | 9783959770415 |

DOIs | |

Publication status | Published - 1 Jul 2017 |

Externally published | Yes |

Event | 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland Duration: 10 Jul 2017 → 14 Jul 2017 Conference number: 44 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Publisher | Dagstuhl |

Volume | 80 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 |
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Abbreviated title | ICALP 2017 |

Country/Territory | Poland |

City | Warsaw |

Period | 10/07/17 → 14/07/17 |

## Keywords

- Generalized Scheduling
- QPTAS
- Unsplittable flows