### Abstract

Original language | English |
---|---|

Pages (from-to) | 249-260 |

Journal | Lecture Notes in Computer Science |

Issue number | 8494 |

DOIs | |

Publication status | Published - 2014 |

Event | 17th Conference on Integer Programming and Combinatorial Optimization - Cham Heidelberg NewYork Dordrecht London Duration: 23 Jun 2014 → 25 Jun 2014 |

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### Bibliographical note

Proceedings title: Integer Programming and Combinatorial OptimizationPublisher: Springer

Place of publication: Cham Heidelberg NewYork Dordrecht London

Editors: J. Lee, J. Vygen

### Cite this

*Lecture Notes in Computer Science*, (8494), 249-260. https://doi.org/10.1007/978-3-319-07557-0_21

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*Lecture Notes in Computer Science*, no. 8494, pp. 249-260. https://doi.org/10.1007/978-3-319-07557-0_21

**Strong LP Formulations for Scheduling Splittable Jobs on Unrelated Machines.** / Correa, J.R.; Marchetti-Spaccamela, A.; Matuschke, J.; Stougie, L.; Svensson, O.; Verdugo, V.; Verschae, J.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - Strong LP Formulations for Scheduling Splittable Jobs on Unrelated Machines

AU - Correa, J.R.

AU - Marchetti-Spaccamela, A.

AU - Matuschke, J.

AU - Stougie, L.

AU - Svensson, O.

AU - Verdugo, V.

AU - Verschae, J.

N1 - Proceedings title: Integer Programming and Combinatorial Optimization Publisher: Springer Place of publication: Cham Heidelberg NewYork Dordrecht London Editors: J. Lee, J. Vygen

PY - 2014

Y1 - 2014

N2 - We study a natural generalization of the problem of minimizing makespan on unrelated machines in which jobs may be split into parts. The different parts of a job can be (simultaneously) processed on different machines, but each part requires a setup time before it can be processed. First we show that a natural adaptation of the seminal approximation algorithm for unrelated machine scheduling [11] yields a 3-approximation algorithm, equal to the integrality gap of the corresponding LP relaxation. Through a stronger LP relaxation, obtained by applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ ≈ 1.618 is the golden ratio. This ratio decreases to 2 in the restricted assignment setting, matching the result for the classic version. Interestingly, we show that our problem cannot be approximated within a factor better than e/e-1 ≈ 1.582 (unless P = NP). This provides some evidence that it is harder than the classic version, which is only known to be inapproximable within a factor 1.5 - ε. Since our 1 + φ bound remains tight when considering the seemingly stronger machine configuration LP, we propose a new job based configuration LP that has an infinite number of variables, one for each possible way a job may be split and processed on the machines. Using convex duality we show that this infinite LP has a finite representation and can be solved in polynomial time to any accuracy, rendering it a promising relaxation for obtaining better algorithms. © 2014 Springer International Publishing Switzerland.

AB - We study a natural generalization of the problem of minimizing makespan on unrelated machines in which jobs may be split into parts. The different parts of a job can be (simultaneously) processed on different machines, but each part requires a setup time before it can be processed. First we show that a natural adaptation of the seminal approximation algorithm for unrelated machine scheduling [11] yields a 3-approximation algorithm, equal to the integrality gap of the corresponding LP relaxation. Through a stronger LP relaxation, obtained by applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ ≈ 1.618 is the golden ratio. This ratio decreases to 2 in the restricted assignment setting, matching the result for the classic version. Interestingly, we show that our problem cannot be approximated within a factor better than e/e-1 ≈ 1.582 (unless P = NP). This provides some evidence that it is harder than the classic version, which is only known to be inapproximable within a factor 1.5 - ε. Since our 1 + φ bound remains tight when considering the seemingly stronger machine configuration LP, we propose a new job based configuration LP that has an infinite number of variables, one for each possible way a job may be split and processed on the machines. Using convex duality we show that this infinite LP has a finite representation and can be solved in polynomial time to any accuracy, rendering it a promising relaxation for obtaining better algorithms. © 2014 Springer International Publishing Switzerland.

U2 - 10.1007/978-3-319-07557-0_21

DO - 10.1007/978-3-319-07557-0_21

M3 - Article

SP - 249

EP - 260

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

IS - 8494

ER -