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A successive approximation algorithm for the multiple knapsack problem

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Abstract

It is well-known that the multiple knapsack problem is NP-hard, and does not admit an FPTAS even for the case of two identical knapsacks. Whereas the 0-1 knapsack problem with only one knapsack has been intensively studied, and some effective exact or approximation algorithms exist. A natural approach for the multiple knapsack problem is to pack the knapsacks successively by using an effective algorithm for the 0-1 knapsack problem. This paper considers such an approximation algorithm that packs the knapsacks in the nondecreasing order of their capacities. We analyze this algorithm for 2 and 3 knapsack problems by the worst-case analysis method and give all their error bounds.

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References

  • Caprara A, Pferschy U (2004) Worst-case analysis of the subset sum algorithm for bin packing. Oper Res Lett 32:159–166

    Article  MATH  MathSciNet  Google Scholar 

  • Caprara A, Kellerer H, Pferschy U (2000a) The multiple subset sum problem. SIAM J Optim 6:308–319

    Article  MathSciNet  Google Scholar 

  • Caprara A, Kellerer H, Pferschy U (2000b) A PTAS for the multiple subset sum problem with different knapsack capacities. Inf Process Lett 73:111–118

    Article  MATH  MathSciNet  Google Scholar 

  • Caprara A, Kellerer H, Pferschy U (2003) A 3/4-approximation algorithm for multiple subset sum. J Heuristics 9:99–111

    Article  MATH  Google Scholar 

  • Chekuri C, Khanna S (2005) A polynomial time approximation scheme for the multiple knapsack problem. SIAM J Comput 35:713–728

    Article  MathSciNet  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco

    MATH  Google Scholar 

  • Horowitz E, Sahni S (1974) Computing partitions with applications to the knapsack problem. J ACM 21:277–292

    Article  MATH  MathSciNet  Google Scholar 

  • Ibarra OH, Kim CE (1975) Fast approximation algorithms for the knapsack and sum of subset problem. J ACM 22:463–468

    Article  MATH  MathSciNet  Google Scholar 

  • Kellerer H, Pferschy U (1999) A new fully polynomial time approximation scheme for the knapsack problem. J Comb Optim 3:59–71

    Article  MATH  MathSciNet  Google Scholar 

  • Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, Berlin

    MATH  Google Scholar 

  • Lawler EL (1979) Fast approximation algorithms for knapsack problems. Math Oper Res 4:339–356

    Article  MATH  MathSciNet  Google Scholar 

  • Martello S, Toth P (1977) An upper bound for the zero-one knapsack problem and a branch and bound algorithm. Eur J Oper Res 1:169–175

    Article  MATH  MathSciNet  Google Scholar 

  • Martello S, Toth P (1981) A branch and bound algorithm for the zero-one multiple knapsack problem. Discrete Appl Math 3:275–288

    Article  MATH  MathSciNet  Google Scholar 

  • Martello S, Pisinger D, Toth P (2000) New trends in exact algorithms for the 0-1 knapsack problem. Eur J Oper Res 123:325–332

    Article  MATH  MathSciNet  Google Scholar 

  • Nauss RM (1976) An efficient algorithm for the 0-1 knapsack problem. Manag Sci 23:27–31

    Article  MATH  Google Scholar 

  • Pisinger D (1999) An exact algorithm for the large multiple knapsack problems. Eur J Oper Res 114:528–541

    Article  MATH  Google Scholar 

  • Pisinger D, Toth P (1998) Knapsack problems. In: Handbook of combinatorial optimization, vol 1. Kluwer Academic, Dordrecht, pp 299–428

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Zhenbo Wang & Wenxun Xing

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  1. Zhenbo Wang
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  2. Wenxun Xing
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Correspondence to Zhenbo Wang.

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Wang, Z., Xing, W. A successive approximation algorithm for the multiple knapsack problem. J Comb Optim 17, 347–366 (2009). https://doi.org/10.1007/s10878-007-9116-y

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