, Volume 10, Issue 2, pp 111–161 | Cite as

The symmetric quadratic knapsack problem: approximation and scheduling applications

  • Hans Kellerer
  • Vitaly A. StrusevichEmail author
Invited Survey


This paper reviews two problems of Boolean non-linear programming: the Symmetric Quadratic Knapsack Problem and the Half-Product Problem. The problems are related since they have a similar quadratic non-separable objective function. For these problems, we focus on the development of fully polynomial-time approximation schemes, especially of those with strongly polynomial time, and on their applications to various scheduling problems.


Quadratic knapsack Half-product Single machine scheduling FPTAS 

MSC classification (2000)

90-02 90C09 90C20 90C59 90B35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adiri I, Bruno J, Frostig E, Rinnooy Kan AHG (1989) Single machine flow-time scheduling with a single breakdown. Acta Inform 26: 679–696CrossRefGoogle Scholar
  2. Agnetis A, Mirchandani PB, Pacciarelli D, Pacifici A (2004) Scheduling problems with two competing agents. Oper Res 52: 229–242CrossRefGoogle Scholar
  3. Badics T, Boros E (1998) Minimization of half-products. Math Oper Res 33: 649–660CrossRefGoogle Scholar
  4. Bagchi U, Sullivan RS, Chang Y-L (1987) Minimizing mean squared deviation of completion times about a common due date. Manag Sci 33: 894–906CrossRefGoogle Scholar
  5. Berman P, Kovoor N, Pardalos PM (1993) Algorithms for the least distance problem. In: Pardalos PM (eds) Complexity in numerical optimization. World Scientific, Singapore, pp 33–56CrossRefGoogle Scholar
  6. Breit J (2007) Improved approximation for non-preemptive single machine flow-time scheduling with an availability constraint. Europ J Oper Res 183: 516–524CrossRefGoogle Scholar
  7. Bretthauer KM, Shetty B (1997) Quadratic resource allocation with generalized upper bounds. Oper Res Lett 20: 51–57CrossRefGoogle Scholar
  8. Brucker P (1984) An O(n) algorithm for quadratic knapsack problems. Oper Res Lett 3: 163–166CrossRefGoogle Scholar
  9. Cai X (1995) Minimization of agreeably weighted variance in single machine systems. Europ J Oper Res 85: 576–592CrossRefGoogle Scholar
  10. Cheng J, Kubiak W (2005) A half-product based approximation scheme for agreeably weighted completion time variance. Europ J Oper Res 162: 45–54CrossRefGoogle Scholar
  11. De P, Ghosh JB, Wells CE (1989) A note on the minimization of mean squared deviation of completion times about a common due date. Manag Sci 35: 1143–1147CrossRefGoogle Scholar
  12. De P, Ghosh JB, Wells CE (1992) On the minimization of completion time variance with bicriteria extension. Oper Res 40: 1148–1155CrossRefGoogle Scholar
  13. Eilon S, Chowdhury IE (1972) Minimizing time variance in the single machine problem. Manag Sci 23: 567–575CrossRefGoogle Scholar
  14. Epstein L, Levin A, Marchetti-Spaccamela A, Megow N, Mestre J, Skutella M, Stogie L (2010) Universal sequencing on a single machine. In: Eisenbrand F, Shepherd B (eds) IPCO 2010, Lect Notes Comp Sci 6080:230–243Google Scholar
  15. Erel E, Ghosh JB (2008) FPTAS for half-products minimization with scheduling applications. Discr Appl Math 156: 3046–3056CrossRefGoogle Scholar
  16. Fathi Y, Nuttle HWL (1990) Heuristics for the common due date weighted tardiness problem. IIE Trans 22: 215–225CrossRefGoogle Scholar
  17. Gordon VS, Potts CN, Strusevich VA, Whitehead JD (2008) Single machine scheduling models with deterioration and learning: handling precedence constraints via priority generation. J Sched 11: 357–370CrossRefGoogle Scholar
  18. Hall NG, Posner ME (1991) Earliness-tardiness scheduling problems, I: weighted deviation of completion times about a common due date. Oper Res 39: 836–846CrossRefGoogle Scholar
  19. Hall NG, Kubiak W, Sethi SP (1991) Earliness-tardiness scheduling problems, II: deviation of completion times about a restrictive common due date. Oper Res 39: 847–856CrossRefGoogle Scholar
  20. Hochbaum DS (2005) Complexity and algorithms for convex network optimization and other nonlinear problems. 4OR Q J Oper Res 3: 171–216CrossRefGoogle Scholar
  21. Hochbaum DS, Shantikumar JG (1990) Convex separable optimization is not much harder than linear optimization. J ACM 37: 843–862CrossRefGoogle Scholar
  22. Hoogeveen JA, Oosterhout H, van de Velde SL (1994) New lower and upper bounds for scheduling around a small common due date. Oper Res 42: 102–110CrossRefGoogle Scholar
  23. Hoogeveen JA, van de Velde SL (1991) Scheduling around a small common due date. Europ J Oper Res 55: 237–242CrossRefGoogle Scholar
  24. Hoogeveen H, Woeginger GJ (2002) Some comments on sequencing with controllable processing times. Computing 68: 181–192CrossRefGoogle Scholar
  25. Janiak A, Kovalyov MY, Kubiak W, Werner F (2005) Positive half-products and scheduling with controllable processing times. Europ J Oper Res 165: 416–422CrossRefGoogle Scholar
  26. Jurisch B, Kubiak W, Józefowska J (1997) Algorithms for minclique scheduling problems. Discr Appl Math 72: 115–139CrossRefGoogle Scholar
  27. Kacem I (2008) Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval. Comp Industr Eng 54: 401–410CrossRefGoogle Scholar
  28. Kacem I (2010) Fully polynomial-time approximation scheme for the weighted total tardiness minimization with a common due date. Discr Appl Math 158: 1035–1040CrossRefGoogle Scholar
  29. Kacem I, Chu C (2008) Worst-case analysis of the WSPT and MWSPT rules for single machine scheduling with one planned setup period. Europ J Oper Res 187: 1080–1089CrossRefGoogle Scholar
  30. Kacem I, Mahjoub AR (2009) Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval. Comp Industr Eng 56: 1708–1712 (see also Erratum: Kacem I (2011) to be published in Comp Industr Eng)Google Scholar
  31. Kanet JT (1981) Minimizing variation of flow time in single machine systems. Manag Sci 27: 1453–1459CrossRefGoogle Scholar
  32. Karakostas G, Kolliopoulos SG, Wang J (2009) An FPTAS for the total weighted tardiness problem with a fixed number of distinct due dates. In: Proceedings of the 15th annual international computing and combinatorics conference (COCOON), Lect Notes Comput Sci 5609:238–248Google Scholar
  33. Kellerer H, Kubzin MA, Strusevich VA (2009) Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval. Europ J Oper Res 199: 111–116CrossRefGoogle Scholar
  34. Kellerer H, Mansini R, Pferschy U, Speranza MG (2003) An efficient fully polynomial approximation scheme for the Subset-Sum Problem. J Comput Syst Sci 66: 349–370CrossRefGoogle Scholar
  35. Kellerer H, Pferschy U (1999) A new fully polynomial time approximation scheme for the knapsack problem. J Combin Optimiz 3: 59–71CrossRefGoogle Scholar
  36. Kellerer H, Pferschy U (2004) Improved dynamic programming in connection with an FPTAS for the knapsack problem. J Combin Optimiz 8: 5–11CrossRefGoogle Scholar
  37. Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, BerlinGoogle Scholar
  38. Kellerer H, Rustogi K, Strusevich VA (2011) Approximation schemes for scheduling on a single machine subject to cumulative deterioration and maintenance. Report SORG-02-2011Google Scholar
  39. Kellerer H, Strusevich VA (2006) A fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date. Theor Comp Sci 369: 230–238CrossRefGoogle Scholar
  40. Kellerer H, Strusevich VA (2010) Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications. Algorithmica 57: 769–795CrossRefGoogle Scholar
  41. Kellerer H, Strusevich VA (2010) Minimizing total weighted earliness-tardiness on a single machine around a small common due date: An FPTAS using quadratic knapsack. Int J Found Comp Sci 21: 357–383CrossRefGoogle Scholar
  42. Kolliopoulos SV, Steiner G (2006) Approximation algorithms for minimizing the total weighted tardiness on a single machine. Theor Comput Sci 355: 261–273CrossRefGoogle Scholar
  43. Kovalyov MY, Kubiak W (1999) A fully polynomial approximation scheme for the weighted earliness-tardiness problem. Oper Res 47: 757–761CrossRefGoogle Scholar
  44. Kozlov MK, Tarasov SP, Hačijan LG (1979) Polynomial solvability of convex quadratic programming. Sov Math Doklady 20: 1108–1111Google Scholar
  45. Kubiak W (1993) Completion time variance on a single machine is difficult. Oper Res Lett 14: 49–59CrossRefGoogle Scholar
  46. Kubiak W (1995) New results on the completion time variance minimization. Discr Appl Math 58: 157–168CrossRefGoogle Scholar
  47. Kubiak W (2005) Minimization of ordered, symmetric half-products. Discr Appl Math 146: 287–300CrossRefGoogle Scholar
  48. Kubiak W, Cheng J, Kovalyov MY (2002) Fast fully polynomial approximation schemes for minimizing completion time variance. Europ J Oper Res 137: 303–309CrossRefGoogle Scholar
  49. Kubzin MA, Strusevich VA (2005) Two-machine flow shop no-wait scheduling with machine maintenance. 4OR – Q J Oper Res 3: 303–313CrossRefGoogle Scholar
  50. Kubzin MA, Strusevich VA (2006) Planning machine maintenance in two-machine shop scheduling. Oper Res 54: 789–800CrossRefGoogle Scholar
  51. Kuo W-H, Yang D-L (2006) Minimizing the makespan in a single machine scheduling problem with a time-based learning effect. Inform Proc Lett 97: 64–67CrossRefGoogle Scholar
  52. Lawler EL, Moore JM (1969) A functional equation and its application to resource allocation and sequencing problems. Manag Sci 16: 77–84CrossRefGoogle Scholar
  53. Lee C-Y (1996) Machine scheduling with an availability constraint. J Glob Optimiz 9: 395–416CrossRefGoogle Scholar
  54. Lee C-Y (2004) Machine scheduling with availability constraints. In: Leung JY-T (eds) Handbook of scheduling: algorithms, models and performance analysis. Chapman and Hall/CRC, London, pp 22-1–22-13Google Scholar
  55. Lee C-Y, Liman SD (1992) Single machine flow time scheduling with scheduled maintenance. Acta Inform 29: 375–382CrossRefGoogle Scholar
  56. Ma Y, Chu C, Zuo C (2010) A survey of scheduling with deterministic machine availability constraints. Comp Industr Eng 58: 199–211CrossRefGoogle Scholar
  57. Marchetti-Spaccamela A, Megow N, Skutella M, Stougie L (2008) Robust sequencing on a single machine. Matheon Preprint 533Google Scholar
  58. Martello S, Toth P (1990) Knapsack Problems. Algorithms and computer implementation. Wiley, ChichesterGoogle Scholar
  59. Merten AG, Muller ME (1972) Variance minimization in single machine sequencing problems. Manag Sci 18: 518–528CrossRefGoogle Scholar
  60. Megow N, Verschae J (2009) Short note on scheduling on a single machine with one non-availability period. Matheon Preprint 557Google Scholar
  61. Monteiro RDC, Adler I (1989) Interior path following primal-dual algorithms. Part II: convex quadratic programming. Math Progr 44: 43–66CrossRefGoogle Scholar
  62. Moré JJ, Vavasis SA (1991) On the solution of concave knapsack problems. Math Progr 49: 397–411CrossRefGoogle Scholar
  63. Nowicki E, Zdrzałka S (1990) A survey of results for sequencing problems with controllable processing times. Discr Appl Math 26: 271–287CrossRefGoogle Scholar
  64. Pisinger D (2007) The quadratic knapsack problem—a survey. Discr Appl Math 155: 623–648CrossRefGoogle Scholar
  65. Rader DJ Jr., Woeginger GJ (2002) The quadratic 0–1 knapsack problem with series–parallel support. Oper Res Lett 30: 159–166CrossRefGoogle Scholar
  66. Romeijn HE, Geunes G, Taafe K (2007) On a nonseparable convex maximization problem with continuous knapsack constraints. Oper Res Lett 35: 172–180CrossRefGoogle Scholar
  67. Sadfi C, Penz B, Rapin C, Błažewicz J, Formanowicz P (2005) An improved approximation algorithm for the single machine total completion time scheduling problem with availability constraints. Europ J Oper Res 161: 3–10CrossRefGoogle Scholar
  68. Sahni SK (1976) Algorithms for scheduling independent tasks. J ACM 23: 116–127CrossRefGoogle Scholar
  69. Shabtay D, Steiner G (2007) A survey of scheduling with controllable processing times. Discr Appl Math 155: 1643–1666CrossRefGoogle Scholar
  70. Shakhlevich NV, Strusevich VA (2006) Single machine scheduling with controllable release and processing parameters. Discr Appl Math 154: 2178–2199CrossRefGoogle Scholar
  71. Skutella M (2001) Convex quadratic and semidefinite programming relaxations in scheduling. J ACM 48: 206–242CrossRefGoogle Scholar
  72. Smith WE (1956) Various optimizers for single stage production. Naval Res Logist Quart 3: 59–66CrossRefGoogle Scholar
  73. Tamir A (1993) A strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on two-terminal series-parallel networks. Math Progr 59: 117–132CrossRefGoogle Scholar
  74. Vickson RG (1980) Two single machine sequencing problems involving controllable job processing time. AIIE Trans 12: 258–262CrossRefGoogle Scholar
  75. Wan G, Yen BP-C, Li C-L (2001) Single machine scheduling to minimize total compression plus weighted flow cost is NP-hard. Inform Proc Lett 79: 273–280CrossRefGoogle Scholar
  76. Wang G, Sun H, Chu C (2005) Preemptive scheduling with availability constraints to minimize total weighted completion times. Ann Oper Res 133: 183–192CrossRefGoogle Scholar
  77. Woeginger GJ (1999) An approximation scheme for minimizing agreeably weighted variance on a single machine. INFORMS J Comput 11: 211–216CrossRefGoogle Scholar
  78. Wu C-C, Yin Y, Cheng S-R (2011) Some single-machine scheduling problems with a truncation learning effect. Comp Industr Eng 60: 790–795CrossRefGoogle Scholar
  79. Yang S-J, Yang D-L (2010) Minimizing the makespan single-machine scheduling with aging effects and variable maintenance activities. Omega 38: 528–533CrossRefGoogle Scholar
  80. Yuan J (1992) The NP-hardness of the single machine common due date weighted tardiness problem. Syst Sci Math Sci 5: 328–333Google Scholar
  81. Zhao C-L, Tang H-Y (2010) Single machine scheduling with general job-dependent aging effect and maintenance activities to minimize makespan. Appl Math Model 34: 837–841CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut für Statistik und Operations ResearchUniversität GrazGrazAustria
  2. 2.School of Computing and Mathematical ScienceUniversity of GreenwichLondonUK

Personalised recommendations