LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times

A. Grigoriev; M. Sviridenko; M.J. Uetz

doi:10.1007/11830924_15

LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times

A. Grigoriev^*, M. Sviridenko, M.J. Uetz

^*Corresponding author for this work

Quantitative Economics

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

Abstract

We consider a scheduling problem on unrelated parallel machines with the objective to minimize the makespan. In addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time. The more of the resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied by Shmoys and Tardos. On the basis of an integer linear programming formulation for (a relaxation of) the problem, we adopt a randomized LP rounding technique from Kumar et al. (FOCS 2005) in order to obtain a deterministic, integral LP solution that is close to optimum. We show how this rounding procedure can be used to derive a deterministic 3.75-approximation algorithm for the scheduling problem. This improves upon previous results, namely a deterministic 6.83-approximation, and a randomized 4-approximation. The improvement is due to the better LP rounding and a new scheduling algorithm that can be viewed as a restricted version of the harmonic algorithm for bin packing.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006
Editors	J. Diaz, K. Jansen, J. Rolim, U. Zwick
Place of Publication	Berlin, Heidelberg
Publisher	Springer, Berlin, Heidelberg
Pages	140-151
ISBN (Electronic)	978-3-540-38045-0
ISBN (Print)	978-3-540-38044-3
DOIs	https://doi.org/10.1007/11830924_15
Publication status	Published - 1 Jan 2006
Event	conference; 2006-01-01; 2006-01-01 - Duration: 1 Jan 2006 → 1 Jan 2006

Publication series

Series	Lecture Notes in Computer Science
Volume	4110
ISSN	0302-9743

Conference

Conference	conference; 2006-01-01; 2006-01-01
Period	1/01/06 → 1/01/06

Access to Document

10.1007/11830924_15

Cite this

Grigoriev, A., Sviridenko, M., & Uetz, M. J. (2006). LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times. In J. Diaz, K. Jansen, J. Rolim, & U. Zwick (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 (pp. 140-151). Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_15

Grigoriev, A. ; Sviridenko, M. ; Uetz, M.J. / LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006. editor / J. Diaz ; K. Jansen ; J. Rolim ; U. Zwick. Berlin, Heidelberg : Springer, Berlin, Heidelberg, 2006. pp. 140-151 (Lecture Notes in Computer Science, Vol. 4110).

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abstract = "We consider a scheduling problem on unrelated parallel machines with the objective to minimize the makespan. In addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time. The more of the resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied by Shmoys and Tardos. On the basis of an integer linear programming formulation for (a relaxation of) the problem, we adopt a randomized LP rounding technique from Kumar et al. (FOCS 2005) in order to obtain a deterministic, integral LP solution that is close to optimum. We show how this rounding procedure can be used to derive a deterministic 3.75-approximation algorithm for the scheduling problem. This improves upon previous results, namely a deterministic 6.83-approximation, and a randomized 4-approximation. The improvement is due to the better LP rounding and a new scheduling algorithm that can be viewed as a restricted version of the harmonic algorithm for bin packing.",

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Grigoriev, A, Sviridenko, M & Uetz, MJ 2006, LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times. in J Diaz, K Jansen, J Rolim & U Zwick (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006. Springer, Berlin, Heidelberg, Berlin, Heidelberg, Lecture Notes in Computer Science, vol. 4110, pp. 140-151, conference; 2006-01-01; 2006-01-01, 1/01/06. https://doi.org/10.1007/11830924_15

LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times. / Grigoriev, A.; Sviridenko, M.; Uetz, M.J.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006. ed. / J. Diaz; K. Jansen; J. Rolim; U. Zwick. Berlin, Heidelberg: Springer, Berlin, Heidelberg, 2006. p. 140-151 (Lecture Notes in Computer Science, Vol. 4110).

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

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N2 - We consider a scheduling problem on unrelated parallel machines with the objective to minimize the makespan. In addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time. The more of the resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied by Shmoys and Tardos. On the basis of an integer linear programming formulation for (a relaxation of) the problem, we adopt a randomized LP rounding technique from Kumar et al. (FOCS 2005) in order to obtain a deterministic, integral LP solution that is close to optimum. We show how this rounding procedure can be used to derive a deterministic 3.75-approximation algorithm for the scheduling problem. This improves upon previous results, namely a deterministic 6.83-approximation, and a randomized 4-approximation. The improvement is due to the better LP rounding and a new scheduling algorithm that can be viewed as a restricted version of the harmonic algorithm for bin packing.

AB - We consider a scheduling problem on unrelated parallel machines with the objective to minimize the makespan. In addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time. The more of the resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied by Shmoys and Tardos. On the basis of an integer linear programming formulation for (a relaxation of) the problem, we adopt a randomized LP rounding technique from Kumar et al. (FOCS 2005) in order to obtain a deterministic, integral LP solution that is close to optimum. We show how this rounding procedure can be used to derive a deterministic 3.75-approximation algorithm for the scheduling problem. This improves upon previous results, namely a deterministic 6.83-approximation, and a randomized 4-approximation. The improvement is due to the better LP rounding and a new scheduling algorithm that can be viewed as a restricted version of the harmonic algorithm for bin packing.

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M3 - Conference article in proceeding

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BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006

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A2 - Zwick, U.

PB - Springer, Berlin, Heidelberg

CY - Berlin, Heidelberg

T2 - conference; 2006-01-01; 2006-01-01

Y2 - 1 January 2006 through 1 January 2006

ER -

Grigoriev A, Sviridenko M, Uetz MJ. LP Rounding and an Almost Harmonic Algorithm for Scheduling with Resource Dependent Processing Times. In Diaz J, Jansen K, Rolim J, Zwick U, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006. Berlin, Heidelberg: Springer, Berlin, Heidelberg. 2006. p. 140-151. (Lecture Notes in Computer Science, Vol. 4110). doi: 10.1007/11830924_15