Approximability of average completion time scheduling on unrelated machines

We show that minimizing the average job completion time on unrelated machines is APX -hard if preemption of jobs is allowed. This provides one of the last missing pieces in the complexity classification of machine scheduling with (weighted) sum of completion times objective. The proof is based on a...

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Published in:	Mathematical programming Vol. 161; no. 1-2; pp. 135 - 158
Main Author:	Sitters, René
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2017 Springer Nature B.V
Subjects:	Algorithms Approximation Calculus of Variations and Optimal Control; Optimization Classification Combinatorics Completion time Employment Full Length Paper Job shops Linear programming Lower bounds Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Preemption Quadratic programming Scheduling Studies Texts Theoretical hardness 68W25 Approximation algorithms Approximation algorithms 90B35 Scheduling theory, deterministic Scheduling 68Q25 Analysis of algorithms and problem complexity Quadratic programming Average completion time
ISSN:	0025-5610, 1436-4646
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Abstract	We show that minimizing the average job completion time on unrelated machines is APX -hard if preemption of jobs is allowed. This provides one of the last missing pieces in the complexity classification of machine scheduling with (weighted) sum of completion times objective. The proof is based on a mixed integer linear program. This means that verification of the reduction is partly done by an ILP-solver. This gives a concise proof which is easy to verify. In addition, we give a deterministic 1.698-approximation algorithm for the weighted version of the problem. The improvement is made by modifying and combining known algorithms and by the use of new lower bounds. These results improve on the known NP -hardness and 2-approximability.
AbstractList	We show that minimizing the average job completion time on unrelated machines is APX -hard if preemption of jobs is allowed. This provides one of the last missing pieces in the complexity classification of machine scheduling with (weighted) sum of completion times objective. The proof is based on a mixed integer linear program. This means that verification of the reduction is partly done by an ILP-solver. This gives a concise proof which is easy to verify. In addition, we give a deterministic 1.698-approximation algorithm for the weighted version of the problem. The improvement is made by modifying and combining known algorithms and by the use of new lower bounds. These results improve on the known NP -hardness and 2-approximability. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We show that minimizing the average job completion time on unrelated machines is ...-hard if preemption of jobs is allowed. This provides one of the last missing pieces in the complexity classification of machine scheduling with (weighted) sum of completion times objective. The proof is based on a mixed integer linear program. This means that verification of the reduction is partly done by an ILP-solver. This gives a concise proof which is easy to verify. In addition, we give a deterministic 1.698-approximation algorithm for the weighted version of the problem. The improvement is made by modifying and combining known algorithms and by the use of new lower bounds. These results improve on the known ...-hardness and 2-approximability.
Author	Sitters, René
Author_xml	– sequence: 1 givenname: René surname: Sitters fullname: Sitters, René email: r.a.sitters@vu.nl, sitters@cwi.nl organization: Vrije Universiteit, Centrum Wiskunde & Informatica (CWI)
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Cites_doi	10.1137/S0097539799358094 10.1016/j.dam.2005.05.014 10.1287/ijoc.13.2.157.10520 10.1007/3-540-48224-5_69 10.1016/S0196-6774(02)00251-1 10.1109/SFFCS.1999.814574 10.1145/361011.361064 10.1145/322092.322101 10.1007/978-3-540-87744-8_64 10.1016/S0167-5060(08)70356-X 10.1137/S089548019936223X 10.1145/321978.321985 10.1287/opre.21.3.846 10.1007/3-540-63248-4_11 10.1137/S0895480199357078 10.1002/(SICI)1099-1425(199909/10)2:5<203::AID-JOS26>3.0.CO;2-5 10.1145/375827.375840 10.1016/j.jalgor.2004.06.011 10.1016/0020-0190(91)90246-E 10.1145/301250.301356
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References	QueyranneMSchulzASApproximation bounds for a general class of precedence constrained parallel machine scheduling problemsSIAM J. Comput.20063551241125310.1137/S00975397993580941100.680102217144 Skutella, M., Woeginger, G.: A PTAS for minimizing the weighted sum of job completion times on parallel machines. In: Proceedings of 31st Symposium Theory of Computing, pp. 400–407 (1999) Goemans, M.: Improved approximation algorithms for scheduling with release dates. In: Proceedings of 8th Symposium on Discrete Algorithms, pp. 591–598. New Orleans, Louisiana, USA (1997) SittersRComplexity of preemptive minsum scheduling on unrelated parallel machinesJ. Algorithms2005571374810.1016/j.jalgor.2004.06.0111101.684302167167 SkutellaMConvex quadratic and semidefinite programming relaxations in schedulingJ. ACM200148220624210.1145/375827.3758401323.900241868715 GrahamRLawlerELenstraJKanAROptimization and approximation in deterministic sequencing and scheduling: a surveyAnn. Discret. Math.1979528732610.1016/S0167-5060(08)70356-X0411.90044558574 KannVMaximum bounded 3-dimensional matching is MAX SNP-completeInf. Process. Lett.199137273510.1016/0020-0190(91)90246-E0711.680451089773 SchuurmanPWoegingerGPolynomial time approximation algorithms for machine scheduling: ten open problemsJ. Sched.1999220321310.1002/(SICI)1099-1425(199909/10)2:5<203::AID-JOS26>3.0.CO;2-50938.900321991779 Afrati, F., Bampis, E., Chekuri, C., Karger, D., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., Sviridenko, M.: Approximation schemes for minimizing average weighted completion time with release dates. In: FOCS ’99, pp. 32–44 (1999) Chekuri, C., Khanna, S.: A PTAS for minimizing weighted completion time on uniformly related machines. In: ICALP. Lecture Notes in Computer Science, vol. 2076, pp. 848–861 (2001) SchulzASkutellaMScheduling unrelated machines by randomized roundingSIAM J. Discret. Math.20021545046910.1137/S08954801993570781055.900401935832 BrunoJCoffmanEGJrSethiRScheduling independent tasks to reduce mean finishing timeCommun. ACM19741738238710.1145/361011.3610640283.68039356869 GonzalezTSahniSOpen shop scheduling to minimize finish timeJ. ACM19762366567910.1145/321978.3219850343.68031429089 Sitters, R.: Approximability of average completion time scheduling on unrelated machines. In: Halperin, D., Mehlhorn, K. (eds.) Proceedings of 16th European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 5193, pp. 768–779. Springer (2008) AfratiFMilisIDesigning PTASs for min-sum scheduling problemsDiscret. Appl. Math.200615462263910.1016/j.dam.2005.05.0141120.900142202469 HoogeveenJSchuurmanPWoegingerGNon-approximability results for scheduling problems with minsum criteriaINFORMS J. Comput.20011315716810.1287/ijoc.13.2.157.105201238.900661904180 GoemansMQueyranneMSchulzASkutellaMWangYSingle machine scheduling with release datesSIAM J. Discret. Math.20021516519210.1137/S089548019936223X1009.900961920180 QueyranneMSviridenkoMA (2+epsilon)-approximation algorithm for the generalized preemptive open shop problem with minsum objectiveJ. Algorithms200245220221210.1016/S0196-6774(02)00251-11030.681051943222 LawlerELabetoulleJOn preemptive scheduling of unrelated parallel processors by linear programmingJ. ACM19782561261910.1145/322092.3221010388.68027508706 Hall, L., Shmoys, D.B., Wein, J.: Scheduling to minimize average completion time: off-line and on-line algorithms. In: Proceedings of 7th Symposium on Discrete Algorithms, pp. 142–151 (1996) Schulz, A., Skutella, M.: Random-based scheduling: new approximations and LP lower bounds. pp. 119–133 (1997) HornWMinimizing average flow time with parallel machinesOper. Res.19732184684710.1287/opre.21.3.8460259.90030 1004_CR19 1004_CR16 M Goemans (1004_CR6) 2002; 15 M Queyranne (1004_CR15) 2002; 45 T Gonzalez (1004_CR7) 1976; 23 J Hoogeveen (1004_CR10) 2001; 13 J Bruno (1004_CR3) 1974; 17 F Afrati (1004_CR2) 2006; 154 1004_CR1 R Graham (1004_CR8) 1979; 5 M Queyranne (1004_CR14) 2006; 35 M Skutella (1004_CR22) 2001; 48 R Sitters (1004_CR20) 2005; 57 1004_CR9 1004_CR21 V Kann (1004_CR12) 1991; 37 E Lawler (1004_CR13) 1978; 25 1004_CR5 A Schulz (1004_CR17) 2002; 15 1004_CR4 W Horn (1004_CR11) 1973; 21 P Schuurman (1004_CR18) 1999; 2
References_xml	– reference: GrahamRLawlerELenstraJKanAROptimization and approximation in deterministic sequencing and scheduling: a surveyAnn. Discret. Math.1979528732610.1016/S0167-5060(08)70356-X0411.90044558574 – reference: SittersRComplexity of preemptive minsum scheduling on unrelated parallel machinesJ. Algorithms2005571374810.1016/j.jalgor.2004.06.0111101.684302167167 – reference: Sitters, R.: Approximability of average completion time scheduling on unrelated machines. In: Halperin, D., Mehlhorn, K. (eds.) Proceedings of 16th European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 5193, pp. 768–779. Springer (2008) – reference: Afrati, F., Bampis, E., Chekuri, C., Karger, D., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., Sviridenko, M.: Approximation schemes for minimizing average weighted completion time with release dates. In: FOCS ’99, pp. 32–44 (1999) – reference: SkutellaMConvex quadratic and semidefinite programming relaxations in schedulingJ. ACM200148220624210.1145/375827.3758401323.900241868715 – reference: AfratiFMilisIDesigning PTASs for min-sum scheduling problemsDiscret. Appl. Math.200615462263910.1016/j.dam.2005.05.0141120.900142202469 – reference: BrunoJCoffmanEGJrSethiRScheduling independent tasks to reduce mean finishing timeCommun. ACM19741738238710.1145/361011.3610640283.68039356869 – reference: QueyranneMSviridenkoMA (2+epsilon)-approximation algorithm for the generalized preemptive open shop problem with minsum objectiveJ. Algorithms200245220221210.1016/S0196-6774(02)00251-11030.681051943222 – reference: Goemans, M.: Improved approximation algorithms for scheduling with release dates. In: Proceedings of 8th Symposium on Discrete Algorithms, pp. 591–598. New Orleans, Louisiana, USA (1997) – reference: HornWMinimizing average flow time with parallel machinesOper. Res.19732184684710.1287/opre.21.3.8460259.90030 – reference: SchulzASkutellaMScheduling unrelated machines by randomized roundingSIAM J. Discret. Math.20021545046910.1137/S08954801993570781055.900401935832 – reference: Hall, L., Shmoys, D.B., Wein, J.: Scheduling to minimize average completion time: off-line and on-line algorithms. In: Proceedings of 7th Symposium on Discrete Algorithms, pp. 142–151 (1996) – reference: HoogeveenJSchuurmanPWoegingerGNon-approximability results for scheduling problems with minsum criteriaINFORMS J. Comput.20011315716810.1287/ijoc.13.2.157.105201238.900661904180 – reference: QueyranneMSchulzASApproximation bounds for a general class of precedence constrained parallel machine scheduling problemsSIAM J. Comput.20063551241125310.1137/S00975397993580941100.680102217144 – reference: Skutella, M., Woeginger, G.: A PTAS for minimizing the weighted sum of job completion times on parallel machines. In: Proceedings of 31st Symposium Theory of Computing, pp. 400–407 (1999) – reference: SchuurmanPWoegingerGPolynomial time approximation algorithms for machine scheduling: ten open problemsJ. Sched.1999220321310.1002/(SICI)1099-1425(199909/10)2:5<203::AID-JOS26>3.0.CO;2-50938.900321991779 – reference: Chekuri, C., Khanna, S.: A PTAS for minimizing weighted completion time on uniformly related machines. In: ICALP. Lecture Notes in Computer Science, vol. 2076, pp. 848–861 (2001) – reference: GonzalezTSahniSOpen shop scheduling to minimize finish timeJ. ACM19762366567910.1145/321978.3219850343.68031429089 – reference: Schulz, A., Skutella, M.: Random-based scheduling: new approximations and LP lower bounds. pp. 119–133 (1997) – reference: KannVMaximum bounded 3-dimensional matching is MAX SNP-completeInf. Process. Lett.199137273510.1016/0020-0190(91)90246-E0711.680451089773 – reference: GoemansMQueyranneMSchulzASkutellaMWangYSingle machine scheduling with release datesSIAM J. Discret. 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Snippet	We show that minimizing the average job completion time on unrelated machines is APX -hard if preemption of jobs is allowed. This provides one of the last... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We show that minimizing the average job completion time on unrelated machines is... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We show that minimizing the average job completion time on unrelated machines is...
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SubjectTerms	Algorithms Approximation Calculus of Variations and Optimal Control; Optimization Classification Combinatorics Completion time Employment Full Length Paper Job shops Linear programming Lower bounds Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Preemption Quadratic programming Scheduling Studies Texts Theoretical
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Title	Approximability of average completion time scheduling on unrelated machines
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