Approximability of Two Variants of Multiple Knapsack Problems

  • Shuichi Miyazaki
  • Naoyuki Morimoto
  • Yasuo Okabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)


This paper considers two variants of Multiple Knapsack Problems. The first one is the Multiple Knapsack Problem with Assignment Restrictions and Capacity Constraints (MK-AR-CC). In the MK-AR-CC(\(k\)) (where \(k\) is a positive integer), a subset of knapsacks is associated with each item and the item can be packed into only those knapsacks (Assignment Restrictions). Furthermore, the size of each knapsack is at least \(k\) times the largest item assignable to the knapsack (Capacity Constraints). The MK-AR-CC(\(k\)) is NP-hard for any constant \(k\). In this paper, we give a polynomial-time \(\left( 1+\frac{2}{k+1}+\epsilon \right) \)-approximation algorithm for the MK-AR-CC(\(k\)), and give a lower bound on the approximation ratio of our algorithm by showing an integrality gap of \(\left( 1+\frac{1}{k}-\epsilon \right) \) for the IP formulation we use in our algorithm, where \(\epsilon \) is an arbitrary small positive constant. The second problem is the Splittable Multiple Knapsack Problem with Assignment Restrictions (S-MK-AR), in which the size of items may exceed the capacity of knapsacks and items can be split and packed into multiple knapsacks. We show that approximating the S-MK-AR with the ratio of \(n^{1-\epsilon }\) is NP-hard even when all the items have the same profit, where \(n\) is the number of items and \(\epsilon \) is an arbitrary positive constant.


Multiple knapsack problem Assignment restrictions Approximation algorithms 


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  1. 1.
    Aerts, J., Korst, J., Spieksma, F.: Approximation of a retrieval problem for parallel disks. In: Proceedings of the 5th Italian Conference on Algorithms and Complexity (CIAC), pp. 178–188 (2003)Google Scholar
  2. 2.
    Aerts, J., Korst, J., Spieksma, F., Verhaegh, W., Woeginger, G.: Random redundant storage in disk arrays: complexity of retrieval problems. IEEE Transactions on Computers 52(9), 1210–1214 (2003)CrossRefGoogle Scholar
  3. 3.
    Cohen, R., Katzir, L., Raz, D.: An efficient approximation for the generalized assignment problem. Information Processing Letters 100(4), 162–166 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dawande, M., Kalagnanam, J., Keskinocak, P., Salman, F.S., Ravi, R.: Approximation algorithms for the multiple knapsack problem with assignment restrictions. Journal of Combinatorial Optimization 4(2), 171–186 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Feige, U., Vondrak, J.: Approximation algorithms for allocation problems: improving the factor of \(1-1/e\). In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 667–676 (2006)Google Scholar
  6. 6.
    Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum separable assignment problems. Mathematics of Operations Research 36(3), 416–431 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: Proceedings of the 16th Annual ACM Symposium on Theory of Computing (STOC), pp. 302–311 (1984)Google Scholar
  8. 8.
    Kellerer, H., Pferschy, U.: A new fully polynomial time approximation scheme for the knapsack problem. Journal of Combinatorial Optimization 3(1), 59–71 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kuhn, H.W.: The hungarian method for the assignment problem. Naval Research Logistics Quarterly 2(1–2), 83–97 (1955)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Magazine, M.J., Oguz, O.: A fully polynomial approximation algorithm for the 0–1 knapsack problem. European Journal of Operational Research 8(3), 270–273 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Murty, K.G.: Network Programming. Prentice-Hall, Inc (1992)Google Scholar
  12. 12.
    Nutov, Z., Beniaminy, I., Yuster, R.: A (\(1-1/e\))-approximation algorithm for the generalized assignment problem. Operations Research Letters 34(3), 283–288 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Sakai, K., Okabe, Y.: Quality-aware energy routing toward on-demand home energy networking. In: Proceeding of the 2011 IEEE Consumer Communications and Networking Conference (CCNC), pp. 1041–1044 (2011)Google Scholar
  14. 14.
    Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Mathematical Programming 62(1–3), 461–474 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Takuno, T., Kitamori, Y., Takahashi, R., Hikihara, T.: AC power routing system in home based on demand and supply utilizing distributed power sources. Energies 4(5), 717–726 (2011)CrossRefGoogle Scholar
  16. 16.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 681–690 (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Shuichi Miyazaki
    • 1
  • Naoyuki Morimoto
    • 2
  • Yasuo Okabe
    • 1
  1. 1.Academic Center for Computing and Media StudiesKyoto UniversityKyotoJapan
  2. 2.Institute for Integrated Cell-Material Sciences (iCeMS)Kyoto UniversitySakyo-kuJapan

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