Skip to main content
Log in

Chapter 2 parallel machines — Efficient algorithms, complexity results

  • Part I Discrete Renewable Resources
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. A.V. Aho, J.E. Hopcroft and J.D. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974).

    Google Scholar 

  2. K.R. Baker,Introduction to Sequencing and Scheduling (J. Wiley & Sons, New York, 1974).

    Google Scholar 

  3. J. Błażewicz, Scheduling dependent tasks with different arrival times to meet deadlines, in: E. Gelenbe, H. Beilner (eds.),Modelling and Performance Evaluation of Computer Systems (North-Holland, Amsterdam, 1976) pp. 57–65.

    Google Scholar 

  4. J. Błażewicz, Simple algorithms for multiprocessor scheduling to meet deadlines, Information Processing Letters 6, No. 3 (1977)162–164.

    Article  Google Scholar 

  5. J. Błażewicz, Scheduling with deadlines and resource constraints, Report, Instytut Automatyki, Politechnika Poznańska, 1977.

  6. J. Błażewicz, Complexity of computer scheduling algorithms under resource constraints, Proc. I Meeting AFCET-SMF on Applied Mathematics, Palaiseau (France), 1978, pp. 169–178.

  7. J. Błażewicz, Scheduling tasks on parallel processors under resource constraints to minimize mean finishing time, Methods of Operations Research, No. 35(1979) pp. 67–72.

  8. J. Błażewicz, Złożoność obliczeniowa algorytmów i problemów szeregowania zadań, Wyd. Politechniki Poznańskiej, Seria Rozprawy, No. 104, 1979.

  9. J. Błażewicz, Deadline scheduling of tasks with ready times and resource constraints, Information Processing Letters 8, No. 2 (1979) pp. 60–63.

    Article  Google Scholar 

  10. J. Błażewicz, Solving the resource constrained deadline scheduling problem via reduction to the network flow problem, European J. of Operational Research 6, No. 1 (1981) 75–79.

    Article  Google Scholar 

  11. J. Błażewicz, Selected topics in scheduling theory, in: G. Laporte, S. Martello, M. Minoux, C. Ribeiro (eds.),Annals of Discrete Mathematics (North Holland, Amsterdam) to appear.

  12. J. Błażewicz, J. Barcelo, W. Kubiak, H. Röck, Scheduling tasks on two processors with deadlines and additional resources, European J. of Operational Res., to appear.

  13. J. Błażewicz, W. Cellary, R. Błowiński and J. Węglarz, Deterministyczne problemy szeregowania zadań na równoległych procesorach. Cz.I. Zbiory zadań niezależnych, Podstawy Sterowania 6, No. 2 (1976) 155–178.

    Google Scholar 

  14. J. Błażewicz, W. Cellary and J. Węglarz, A strategy for scheduling splittable tasks to reduce schedule length, Acta Cybernetica 3, No. 2 (1977)99–106.

    Google Scholar 

  15. J. Błażewicz and K. Ecker, A linear time algorithm for restricted bin packing and scheduling problems, Operations Research Letters 2(1983)80–83.

    Article  Google Scholar 

  16. J. Błażewicz, W. Kubiak, H. Röck and J. Szwarcfiter, Minimizing mean flow time under resource constraints on parallel processors (1986) to appear.

  17. J. Błażewicz, J.K. Lenstra and A.H.G. Rinnooy Kan, Scheduling subject to resource constraints; classification and complexity, Discrete Applied Mathematics 5, No. 1 (1983)11–24.

    Google Scholar 

  18. P.J. Brucker, Sequencing unit-time jobs with treelike precedence on machines to minimize maximum lateness, Proc. IX International Symposium on Mathematical Programming, Budapest, 1976.

  19. P. Brucker, M.R. Garey and D.S. Johnson, Scheduling equal-length tasks under treelike precedence constraints to minimize maximum lateness, Mathematics of Operations Research 2(1977)275–284.

    Google Scholar 

  20. J. Bruno, E.G. Coffman, Jr. and R. Sethi, Scheduling independent tasks to reduce mean finishing time, Comm. ACM 17, No. 7 (1974)382–387.

    Article  Google Scholar 

  21. J. Brzeziński, Preemptive scheduling of processors with private memories, Foundations of Control Engineering 5, No. 2 (1980)55–69.

    Google Scholar 

  22. E.G. Jr. Coffman (ed.),Computer & Job/Shop Scheduling Theory (J. Wiley, New York, 1976).

    Google Scholar 

  23. E.G. Jr. Coffman and R.L. Graham, Optimal scheduling for two processors systems, Acta Informatica 1, No. 3 (1972)200–213.

    Article  Google Scholar 

  24. R.W. Conway, W.L. Maxwell and L.W. Miller,Theory of Scheduling (Addison-Wesley, Reading, Mass, 1967).

    Google Scholar 

  25. D.F. Cooper, A note on serial and parallel heuristic for resource-constrained project scheduling, Foundations of Control Engineering 2, No. 2 (1977)131–134.

    Google Scholar 

  26. E.W. Davis, Project scheduling under resource constraints; historical review and categorization of procedures, AIIE Transactions 5(1973)297–313.

    Google Scholar 

  27. S. Even,Graph Algorithms (Computer Science Press Inc, 1979).

  28. A. Federgruen and H. Groenevelt, Preemptive scheduling of uniform machines by ordinary network flow techniques, to appear.

  29. L.R. Ford, Jr. and D.R. Fulkerson,Flows in Networks (Princeton, New Jersey, 1962).

    Google Scholar 

  30. P. Gaćs and L. Lovasz, Khachian's algorithm for linear programming, Report, Computer Science Department, Stanford University, 1979.

  31. Z. Galil and A. Naamad, Network flow and generalized path compression, Proc. ACM Symp. Theory of Comput. 1979, pp. 13–26.

  32. M.R. Garey, unpublished result.

  33. M.R. Garey and R.L. Graham, Bounds for multiprocessor scheduling with resource constraints, SIAM J. on Computing 4(1975)187–200.

    Article  Google Scholar 

  34. M.R. Garey, R.L. Graham, D.S. Johnson and A.C.-C. Yao, Resource constrained scheduling as generalized bin packing, J. Combinatorial Theory Ser. A 21(1976)257–298.

    Article  Google Scholar 

  35. M.R. Garey and D.S. Johnson, Complexity results for multiprocessor scheduling under resource constraints, SIAM J. on Computing 4(1975)397–411.

    Article  Google Scholar 

  36. M.R. Garey and D.S. Johnson, Scheduling tasks with nonuniform deadlines on two processors, J. ACM 23, No. 3 (1976)461–467.

    Article  Google Scholar 

  37. M.R. Garey and D.S. Johnson, Two-processor scheduling with start times and deadlines, SIAM J. on Computing 6(1977)416–426.

    Article  Google Scholar 

  38. M.R. Garey and D.S. Johnson, ‘Strong’ NP-completeness results: motivation, examples and implication, J. ACM 25, No. 4 (1978)499–508.

    Article  Google Scholar 

  39. M.R. Garey and D.S. Johnson, Approximation algorithms for combinatorial problems: an annotated bibliography, in: J.F. Traub (ed.),Algorithms and Complexity: New Directions and Recent Results (Academic Press, New York, 1976) pp. 41–52.

    Google Scholar 

  40. M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman, San Francisco, 1979).

    Google Scholar 

  41. M.R. Garey, D.S. Johnson, B.B. Simons and R.E. Tarjan, Scheduling unit-time tasks with arbitrary release times and deadlines, SIAM J. Comput. 10(1981)256–269.

    Article  Google Scholar 

  42. T. Gonzalez, Optimal mean finish time preemptive schedules, Technical Report 220, Computer Science Department, Pennsylvania State University, 1977.

  43. T. Gonzalez and D.B. Johnson, A new algorithm for preemptive scheduling of trees, J. ACM 27(1980)287–312.

    Article  Google Scholar 

  44. T. Gonzalez, E.L. Lawler and S. Sahni, Optimal preemptive scheduling of a fixed number of unrelated processors in linear time, to appear.

  45. T. Gonzalez and S. Sahni, Preemptive scheduling of uniform processor systems, J. ACM 25, No. 1 (1978)91–101.

    Article  Google Scholar 

  46. R.L. Graham, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling theory: a survey, Ann. Discrete Math. 5(1979)287–326.

    MathSciNet  Google Scholar 

  47. W.A. Horn, Single — machine job sequencing with treelike precedence ordering and linear delay penalties, SIAM J. on Appl. Math. 23(1972)189–202.

    Article  Google Scholar 

  48. W.A. Horn, Minimizing average flow time with parallel machines, Operations Research 21(1973)846–847.

    Google Scholar 

  49. W.A. Horn, Some simple scheduling algorithms, Naval Res. Logist, Quart. 21(1974)177–185.

    Google Scholar 

  50. E. Horowitz and S. Sahni, Exact and approximate algorithms for scheduling non-identical processors, J. ACM 23(1976)317–327.

    Article  Google Scholar 

  51. T.C. Hu, Parallel sequencing and assembly line problems, Operations Research 9, No. 6 (1961)841–848.

    Google Scholar 

  52. J.R. Jackson, Scheduling a production line to minimize maximum tardiness, Research Report 43, Management Science Research Project, University of California, Los Angeles.

  53. D.G. Kafura and V.Y. Shen, Task scheduling on a multiprocessor system with independent memories, SIAM J. Comput. 6(1977)167–187.

    Article  Google Scholar 

  54. O. Kariv and S. Even, An O(n 2.5) algorithm for maximum matching in general graphs, 16-th Annual Symp. on Foundations of Computer science, IEEE (1975)100–112.

  55. R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher (eds.)Complexity of Computer Computation (Plenum Press, New York, 1972) pp. 85–104.

    Google Scholar 

  56. R.M. Karp, J.K. Lenstra, C.J.H. McDiarmid and A.H.G. Rinnooy Kan, Probabilistic analysis of combinatorial algorithms: an annotated bibliography, in: M. O'hEigeartaigh, J.K. Lenstra and A.H.G. Rinnooy Kan (eds.),Combinatorial Optimization: Annotated Bibliographies (J. Wiley, Chichester, 1984).

    Google Scholar 

  57. A.W. Karzanov, Determining the maximal flow in a network by the method of preflows, Soviet Math. Dokl. 15(1974)434–437.

    Google Scholar 

  58. L.G. Khachian, A polynomial algorithm for linear programming (in Russian), Doklady Akad. Nauk USSR 244(1979)1093–1096.

    Google Scholar 

  59. K.L. Krause, V.Y. Shen and H.D. Schwetman, Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems, J. ACM 22(1975)522–550, Erratum: J. ACM 24(1977)527.

    Article  Google Scholar 

  60. J. Labetoulle, Some theorems on real time scheduling, in: E. Gelenbe and R. Mahl (eds.)Computer Architecture and Networks, (North-Holland, Amsterdam, 1974) pp. 285–298.

    Google Scholar 

  61. J. Labetoulle, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, Preemptive scheduling of uniform machines subject to release dates, in:Progress in Combinatorial Optimization (Academic Press, Toronto, 1984) pp. 245–261.

    Google Scholar 

  62. B.J. Lageweg, unpublished.

  63. B.J. Lageweg, J.K. Lenstra and A.H.G. Rinnooy Kan, Minimizing maximum lateness on one machine: computational experience and some applications, Statistica Neerlandica 30, No. 1 (1976)25–41.

    Google Scholar 

  64. T.H. Lai and S. Sahni, Preemptive scheduling of multiprocessor system with memories to minimizeL max, Technical Report 81-20, Comput. Sci. Dept., University of Minnesota, 1981.

  65. E.L. Lawler, Optimal sequencing of a single machine subject to precedence constraints, Management Sci, 19(1973)544–546.

    Google Scholar 

  66. E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).

    Google Scholar 

  67. E.L. Lawler, Sequencing jobs to minimize total weighted completion time subject to precedence constraints, Ann. Discrete Math. 2 (1978) 75–90.

    Google Scholar 

  68. E.L. Lawler, Preemptive scheduling of precedence constrained jobs on parallel machines, in: M.A.H. Dempster, J.K. Lenstra and A.H.G. Rinnooy Kan (eds.),Deterministic and Stochastic Scheduling (Reidel, Dordrecht, 1982) pp. 101–123.

    Google Scholar 

  69. E.L. Lawler, Recent results in the theory of machine scheduling in: A. Bachem, M. Grötschel and B. Korte (eds.),Mathematical Programming: The State of the Art-Bonn 1982 (Springer Verlag, Berlin, 1982) pp. 202–234.

    Google Scholar 

  70. E.L. Lawler, unpublished.

  71. E.L. Lawler, J. Labetoulle, Preemptive scheduling of unrelated parallel processors by linear programming, J. ACM 25 (1978) 512–619.

    Article  Google Scholar 

  72. E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, Recent developments in deterministic sequencing and scheduling: a survey in: M.A.H. Dempster, J.K. Lenstra, A.H.G. Rinnooy Kan (eds.)Deterministic and Stochastic Scheduling (Reidel, Dordrecht, 1982) pp. 35–73.

    Google Scholar 

  73. E.L. Lawler and J.M. Moors, A functional equation and its application to resource allocation and scheduling problems, Management Sci. 16, No. 1 (1969), 77–84.

    Google Scholar 

  74. H.W. Lenstra, Jr., Integer programming with a fixed number of variables, Report, University of Amsterdam, 1981.

  75. J.K. Lenstra, unpublished.

  76. J.K. Lenstra, Sequencing by Enumerative Methods, Mathematical Centre Tracts 69, Mathematisch Centrum, Amsterdam, 1976.

    Google Scholar 

  77. J.K. Lenstra and A.H.G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Operations Res. 25 (1978), 22–35.

    Google Scholar 

  78. J.K. Lenstra and A.H.G. Rinnooy Kan, Scheduling theory since 1981: an annotated bibliography, in: M. O'hEigeartaigh, J.K. Lenstra, A.H.G. Rinnooy Kan (eds.),Combinatorial Optimization: Annotated Bibliographies (J. Wiley, Chichester, 1984).

    Google Scholar 

  79. J.K. Lenstra, A.H.G. Rinnooy Kan and P. Brucker, Complexity of machine scheduling problems, Ann. Discrete Math. 1(1977)343–362.

    Google Scholar 

  80. J.Y.-T. Leung, Bounds on list scheduling of UET tasks with restricted resource constraints, Information Processing Letters 9, No. 4 (1979)167–170.

    Article  Google Scholar 

  81. C.L. Liu and J.W. Layland, Scheduling algorithms for multiprogramming in a hard-real-time environment, J. ACM 20, No. 1 (1973)46–61.

    Article  Google Scholar 

  82. E.L. Lloyd, Scheduling task systems with resources, MIT Laboratory for Computer Science, Tech. Report No. 236, 1980.

  83. E.L. Lloyd, List scheduling bounds for UET system with resources, Information Processing Letters 10, No. 1 (1980)28–31.

    Article  Google Scholar 

  84. E.L. Lloyd, Coffman-Graham scheduling of UET task systems with 0–1 resources, Information Processing Letters 12, No. 1 (1981) pp. 40–45.

    Article  Google Scholar 

  85. R. McNaughton, Scheduling with deadlines and loss functions, Management Sci. 12, No. 1 (1959)1–12.

    Google Scholar 

  86. R.R. Muntz and E.G. Coffman, Jr., Optimal preemptive scheduling on two-processor systems, IEEE Trans. Computers C-18, No 11 (1969) 1014–1020.

    Google Scholar 

  87. R.R. Muntz and E.G. Coffman, Jr., Optimal preemptive scheduling of real time tasks on multiprocessor systems, J. ACM 17, No. 2 (1970)324–328.

    Article  Google Scholar 

  88. A.H.G. Rinnooy Kan, Private communication.

  89. A.H.G. Rinnooy Kan,Machine Scheduling Problems; Classification Complexity and Computations (Nijhoff, The Hague, 1976).

    Google Scholar 

  90. A.H.G. Rinnooy Kan, Probabilistic analysis of approximation algorithms, in: G. Laporte, S. Martello, M. Minoux, C. Ribeiro, (eds.)Annals of Discrete Mathematics (North-Holland, Amsterdam) to appear.

  91. L. Rosenfeld, unpublished result.

  92. S. Sahni and Y. Cho, Nearly on line scheduling of uniform processor system with release times, SIAM J. Comput. 8(1979)275–285.

    Article  Google Scholar 

  93. S. Sahni and Y. Cho, Scheduling independent tasks with due times on a uniform processor system, J. ACM 27(1980)550–563.

    Article  Google Scholar 

  94. S. Sahni and T. Gonzalez, P-complete approximation problems, J. ACM 23(1976)555–565.

    Article  Google Scholar 

  95. T.J. Schaefer, Private communication to M.R. Garey and D.S. Johnson, 1974.

  96. R. Sethi, On the complexity of mean flow time scheduling, Math. Oper. Res. 2(1977)320–330.

    Google Scholar 

  97. J.B. Sidney, Decomposition algorithms for single-machine sequencing with precedence relations and deferral costs, Operations Res. 23(1975)283–298.

    Google Scholar 

  98. E.A. Silver, R.V. Vidal and D. de Werra, A tutorial on heuristic methods, European J. of Oper. Res. 5(1980)153–162.

    Article  Google Scholar 

  99. B. Simons, A fast algorithm for single processor scheduling, Proc. 19th Ann. Symp. on Foundations of Computer Science, IEEE Computer Society, Long Beach, CA, 1978, pp. 246–252.

    Google Scholar 

  100. B. Simons, A fast algorithm for multiprocessor scheduling Proc. 21st Annual IEEE Symp. Foundations of Computer Science, 1980, pp. 50–53.

  101. L. Słomiński, Probabilistic analysis of combinatorial algorithm a bibliography with selected annotation, Computing 28(1982)257–267.

    Google Scholar 

  102. R. Słowiński, Two approaches to problems of resource allocation among project activities — a comparative study. J. Opl. Res. Soc. 31, No. 8 (1980)711–723.

    Google Scholar 

  103. R. Słowiński, Allocation de resources limitées parmi des tâches exécutées par un ensemble de machines indépendantes, in: M. Pelegrin and J. Delmas (eds.),Comparison of Automatic Control and Operational Research Techniques Applied to Large Systems Analysis and Control (Pergamon Press, Oxford, 1979) pp. 189–195.

    Google Scholar 

  104. R. Słowiński, Optimal and heuristic procedures for project scheduling with multiple constrained resource — a survey, Foundations of Control Engineering 2, No. 1 (1977)33–40.

    Google Scholar 

  105. R. Słowiński, L'ordonnancement des tâches préemtives sur les procedures indépendants en présence de ressources supplémentaires, R.A.I.R.O. Informatique Computer Science 15, No. 2 (1981)155–166.

    Google Scholar 

  106. J.D. Ullman, NP-complete scheduling problems, J. Comput. System Sci. 10(1975)384–393.

    Google Scholar 

  107. J.D. Ullman, Complexity of sequencing problems, chapter 4, in [22], 1976.

  108. J. Węglarz, J. Błażewicz, W. Cellary and R. Słowiński, An automatic revised simplex method for constrained resource network scheduling, ACM Trans. on Mathematical Software 3, No. 3 (1977)295–300.

    Article  Google Scholar 

  109. D. de Werra, Preemptive scheduling, linear programming and network flows, SIAM J. Algebraic Discrete Math. 5(1984)11–20.

    Google Scholar 

  110. W.A. Wulf and C.G. Bell, C.mmp- a multi-mini-processor, Proc. FJJC (1972)766–778.

Download references

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chapter 2 parallel machines — Efficient algorithms, complexity results. Ann Oper Res 7, 9–98 (1986). https://doi.org/10.1007/BF02186434

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02186434

Keywords

Navigation