Journal of Global Optimization

, Volume 69, Issue 3, pp 713–725 | Cite as

Knapsack with variable weights satisfying linear constraints

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Abstract

We introduce a variant of the knapsack problem, in which the weights of items are also variables but must satisfy a system of linear constraints, and the capacity of knapsack is given and known. We discuss two models: (1) the value of each item is given; (2) the value-to-weight ratio of each item is given. The goal is to determine the weight of each item, and to find a subset of items such that the total weight is no more than the capacity and the total value is maximized. We provide several practical application scenarios that motivate our study, and then investigate the computational complexity and corresponding algorithms. In particular, we show that if the number of constraints is a fixed constant, then both problems can be solved in polynomial time. If the number of constraints is an input, then we show that the first problem is NP-Hard and cannot be approximated within any constant factor unless \(\mathrm{P}= \mathrm{NP}\), while the second problem is NP-Hard but admits a polynomial time approximation scheme. We further propose approximation algorithms for both problems, and extend the results to multiple knapsack cases with a fixed number of knapsacks and identical capacities.

Keywords

Knapsack Linear programming Computational complexity Approximation algorithm 

References

  1. 1.
    Adams, W.P., Sherali, H.D.: Mixed-integer bilinear programming problems. Math. Program. 59(1), 279–305 (1993)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bhalgat, A., Goel, A., Khanna, S.: Improved approximation results for stochastic knapsack problems. In: Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’11, pp. 1647–1665. SIAM (2011)Google Scholar
  4. 4.
    Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: A survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)MathSciNetGoogle Scholar
  5. 5.
    Chekuri, C., Khanna, S.: A polynomial time approximation scheme for the multiple knapsack problem. SIAM J. Comput. 35(3), 713–728 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dean, B.C.: Approximation algorithms for stochastic scheduling problems. Ph.D. thesis, Massachusetts Institute of Technology, Boston (2005)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., San Francisco (1979)MATHGoogle Scholar
  8. 8.
    Gupte, A., Ahmed, S., Cheon, M., Dey, S.: Solving mixed integer bilinear problems using MILP formulations. SIAM J. Optim. 23(2), 721–744 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Håstad, J.: Clique is hard to approximate within \(n^{1 - \epsilon }\). In: Proceedings 37th Annual Symposium on Foundations of Computer Science, pp. 627–636 (1996)Google Scholar
  10. 10.
    Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22(4), 463–468 (1975)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, New York (2004)CrossRefMATHGoogle Scholar
  12. 12.
    Köppe, M.: On the complexity of nonlinear mixed-integer optimization. In: Lee, J., Leyffer, S. (eds.) Mixed-Integer Nonlinear Programming. IMA Volumes in Mathematics and its Applications, vol. 154, pp. 533–558. Springer, Berlin (2011)CrossRefGoogle Scholar
  13. 13.
    Monaci, M., Pferschy, U.: On the robust knapsack problem. SIAM J. Optim. 23(4), 1956–1982 (2013)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Nip, K., Wang, Z., Wang, Z.: Scheduling under linear constraints. Eur. J. Oper. Res. 253(2), 290–297 (2016)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Sahni, S.: Algorithms for scheduling independent tasks. J. ACM 23(1), 116–127 (1976)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Wang, Z., Xing, W.: A successive approximation algorithm for the multiple knapsack problem. J. Comb. Optim. 17(4), 347–366 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Ye, Y.: Interior Point Algorithms: Theory and Analysis. Wiley-Interscience, New York (1997)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Department of Industrial and Systems EngineeringUniversity of MinnesotaMinneapolisUSA

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