An efficient polynomial time approximation scheme for load balancing on uniformly related machines

We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, C i denotes the completion time of machine i . Our goal is to find a schedule that minimizes or maximizes ∑...

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Veröffentlicht in:	Mathematical programming Jg. 147; H. 1-2; S. 1 - 23
Hauptverfasser:	Epstein, Leah, Levin, Asaf
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2014 Springer Nature B.V
Schlagworte:	Algorithms Analysis Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Completion time Computer science Dynamic programming Employment Full Length Paper Intervals Job shops Load Load balancing Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Partitions Polynomials Schedules Scheduling Studies Theoretical 90C59 Approximation methods and heuristics 68W25 Approximation algorithms 90C27 Combinatorial optimization Approximation algorithms 68W40 Analysis of algorithms EPTAS Scheduling 68Q25 Analysis of algorithms and problem complexity Load balancing
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Abstract	We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, C i denotes the completion time of machine i . Our goal is to find a schedule that minimizes or maximizes ∑ i = 1 m C i p for a fixed value of p such that 0 < p < ∞ . For p > 1 the minimization problem is equivalent to the well-known problem of minimizing the ℓ p norm of the vector of the completion times of the machines, and for 0 < p < 1 , the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS’s for this natural class of load balancing problems.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, ... denotes the completion time of machine i. Our goal is to find a schedule that minimizes or maximizes ... for a fixed value of p such that ... For ... the minimization problem is equivalent to the well-known problem of minimizing the ... norm of the vector of the completion times of the machines, and for ..., the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS's for this natural class of load balancing problems. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, ... denotes the completion time of machine i. Our goal is to find a schedule that minimizes or maximizes ... for a fixed value of p such that ... For ... the minimization problem is equivalent to the well-known problem of minimizing the ... norm of the vector of the completion times of the machines, and for ..., the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS's for this natural class of load balancing problems.[PUBLICATION ABSTRACT] We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, C i denotes the completion time of machine i . Our goal is to find a schedule that minimizes or maximizes ∑ i = 1 m C i p for a fixed value of p such that 0 < p < ∞ . For p > 1 the minimization problem is equivalent to the well-known problem of minimizing the ℓ p norm of the vector of the completion times of the machines, and for 0 < p < 1 , the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS’s for this natural class of load balancing problems.
Author	Epstein, Leah Levin, Asaf
Author_xml	– sequence: 1 givenname: Leah surname: Epstein fullname: Epstein, Leah email: lea@math.haifa.ac.il organization: Department of Mathematics, University of Haifa – sequence: 2 givenname: Asaf surname: Levin fullname: Levin, Asaf organization: Faculty of Industrial Engineering and Management, The Technion
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Cites_doi	10.1093/comjnl/bxm048 10.1007/978-3-642-18318-8 10.1145/7531.7535 10.1137/090749451 10.1007/s00453-003-1077-7 10.1137/090772228 10.1145/2455.214106 10.1137/0206013 10.1137/0204021 10.1287/moor.6.1.74 10.1016/j.jalgor.2004.02.003 10.1145/321921.321933 10.1145/1132516.1132522 10.1002/j.1538-7305.1966.tb01709.x 10.1007/s004530010051 10.1016/S0020-0190(97)00164-6 10.1016/S0167-6377(96)00055-7 10.1137/0217033 10.1002/(SICI)1099-1425(199806)1:1<55::AID-JOS2>3.0.CO;2-J 10.1145/321941.321951 10.1007/BFb0053962 10.1007/978-1-4612-0515-9 10.1016/j.tcs.2009.08.032 10.1016/j.tcs.2006.05.025 10.1016/0020-0190(95)00099-X 10.1145/800061.808749
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Keywords	90C59 Approximation methods and heuristics 68W25 Approximation algorithms 90C27 Combinatorial optimization Approximation algorithms 68W40 Analysis of algorithms EPTAS Scheduling 68Q25 Analysis of algorithms and problem complexity Load balancing
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SubjectTerms	Algorithms Analysis Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Completion time Computer science Dynamic programming Employment Full Length Paper Intervals Job shops Load Load balancing Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Partitions Polynomials Schedules Scheduling Studies Theoretical
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