Approximation schemes for r-weighted Minimization Knapsack problems

  • Khaled Elbassioni
  • Areg Karapetyan
  • Trung Thanh NguyenEmail author
Original Research


Stimulated by salient applications arising from power systems, this paper studies a class of non-linear Knapsack problems with non-separable quadratic constrains, formulated in either binary or integer form. These problems resemble the duals of the corresponding variants of 2-weighted Knapsack problem (a.k.a., complex-demand Knapsack problem) which has been studied in the extant literature under the paradigm of smart grids. Nevertheless, the employed techniques resulting in a polynomial-time approximation scheme (PTAS) for the 2-weighted Knapsack problem are not amenable to its minimization version. We instead propose a greedy geometry-based approach that arrives at a quasi PTAS (QPTAS) for the minimization variant with boolean variables. As for the integer formulation, a linear programming-based method is developed that obtains a PTAS. In view of the curse of dimensionality, fast greedy heuristic algorithms are presented, additionally to QPTAS. Their performance is corroborated extensively by empirical simulations under diverse settings and scenarios.


Weighted Minimization Knapsack Quasi polynomial-time approximation scheme Polynomial-time approximation scheme Power generation planning Smart grid Economic dispatch control 



We would like to thank the Editor and anonymous reviewers for their careful reading of our manuscript and their helpful comments that improved the presentation of the paper.


  1. Basu, S. (1999). New results on quantifier elimination over real closed fields and applications to constraint databases. Journal of the ACM, 46(4), 537–555.CrossRefGoogle Scholar
  2. Bretthauer, K. M., & Shetty, B. (2002). The nonlinear knapsack problem—Algorithms and applications. European Journal of Operational Research, 138(3), 459–472.CrossRefGoogle Scholar
  3. Chandra, A. K., Hirschberg, D. S., & Wong, C. K. (1976). Approximate algorithms for some generalized knapsack problems. Theoretical Computer Science, 3(3), 293–304.CrossRefGoogle Scholar
  4. Chau, C., Elbassioni, K. M., & Khonji, M. (2016). Truthful mechanisms for combinatorial allocation of electric power in alternating current electric systems for smart grid. ACM Transactions on Economics and Computation, 5(1), 7:1–7:29.CrossRefGoogle Scholar
  5. Chau, S. C., Elbassioni, K. M., & Khonji, M. (2014). Truthful mechanisms for combinatorial AC electric power allocation. In International conference on autonomous agents and multi-agent systems, AAMAS ’14 (pp. 1005–1012), Paris, France, 5–9 May 2014.Google Scholar
  6. Csirik, J., Frenk, J. B. G., Labbé, M., & Zhang, S. (1991). Heuristic for the 0–1 min-knapsack problem. Acta Cybernetica, 10(1–2), 15–20.Google Scholar
  7. Elbassioni, K. M., & Nguyen, T. T. (2017). Approximation algorithms for binary packing problems with quadratic constraints of low cp-rank decompositions. Discrete Applied Mathematics, 230, 56–70.CrossRefGoogle Scholar
  8. Garey, M., & Johnson, D. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: W.H. Freeman.Google Scholar
  9. Hedengren, J. D. (2014). APMonitor Modeling Language. Accessed 18 Aug 2017.
  10. Ibaraki, T., & Katoh, N. (1988). Resource allocation problems. Cambridge, MA: MIT Press.Google Scholar
  11. Karapetyan, A., Khonji, M., Chau, C. K., Elbassioni, K., & Zeineldin, H. (2018). Efficient algorithm for scalable event-based demand response management in microgrids. IEEE Transactions on Smart Grid, 9(4), 2714–2725. Scholar
  12. Kellerer, H., Pferschy, U., & Pisinger, D. (2004). Knapsack problems. Berlin: Springer.CrossRefGoogle Scholar
  13. Kellerer, H., & Strusevich, V. A. (2010). Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications. Algorithmica, 57(4), 769–795.CrossRefGoogle Scholar
  14. Kellerer, H., & Strusevich, V. A. (2012). The symmetric quadratic knapsack problem: approximation and scheduling applications. 4OR, 10(2), 111–161.CrossRefGoogle Scholar
  15. Khonji, M., Karapetyan, A., Elbassioni, K., & Chau, C. K. (2016). Complex-demand scheduling problem with application in smart grid. In Computing and combinatorics (pp. 496–509). Berlin: Springer.Google Scholar
  16. Nemhauser, G. L., & Wolsey, L. A. (1999). Integer and combinatorial optimization. New York, NY: Wiley-Interscience.Google Scholar
  17. Pferschy, U., & Schauer, J. (2013). Approximating the quadratic knapsack problem on special graph classes. In Approximation and online algorithms—11th international workshop, WAOA 2013 (pp. 61–72), Sophia Antipolis, France, September 5–6, 2013, Revised selected papers.Google Scholar
  18. Renegar, J. (1992). On the computational complexity of approximating solutions for real algebraic formulae. SIAM Journal on Computing, 21(6), 1008–1025.CrossRefGoogle Scholar
  19. Schrijver, A. (1986). Theory of linear and integer programming. New York: Wiley.Google Scholar
  20. Rader, D. J, Jr., & Woeginger, G. J. (2002). The quadratic 0–1 knapsack problem with series-parallel support. Operations Research Letters, 30(3), 159–166.CrossRefGoogle Scholar
  21. Woeginger, G. J. (2000). When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS Journal on Computing, 12(1), 57–74.CrossRefGoogle Scholar
  22. Wood, A. J., & Wollenberg, B. F. (2012). Power generation, operation, and control. London: Wiley.Google Scholar
  23. Yu, L., & Chau, C. (2013). Complex-demand knapsack problems and incentives in AC power systems. In International conference on autonomous agents and multi-agent systems, AAMAS ’13 (pp. 973–980), Saint Paul, MN, USA, May 6–10, 2013.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Khaled Elbassioni
    • 1
  • Areg Karapetyan
    • 1
  • Trung Thanh Nguyen
    • 2
    Email author
  1. 1.Masdar Institute, Khalifa University of Science and TechnologyAbu DhabiUAE
  2. 2.Hai Phong UniversityHaiphongVietnam

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