Online Knapsack Problem Under Concave Functions

  • Xin HanEmail author
  • Ning Ma
  • Kazuhisa Makino
  • He Chen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10336)


In this paper, we address an online knapsack problem under concave function f(x), i.e., an item with size x has its profit f(x). We first obtain a simple lower bound \(\max \{q, \frac{f'(0)}{f(1)}\}\), where \(q \approx 1.618\), then show that this bound is not tight, and give an improved lower bound. Finally, we find the online algorithm for linear function [8] can be employed to the concave case, and prove its competitive ratio is \(\frac{f'(0)}{f(1/q)}\), then we give a refined online algorithm with a competitive ratio \(\frac{f'(0)}{f(1)} +1\). And we also give optimal algorithms for some piecewise linear functions.



This research was partially supported by NSFC (11101065), RGC (HKU716412E).


  1. 1.
    Cygan, M., Jez, L., Sgall, J.: Online knapsack revisited. Theory Comput. Syst. 58(1), 153–190 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Han, X., Kawase, Y., Makino, K.: Online unweighted knapsack problem with removal cost. Algorithmica 70(1), 76–91 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Han, X., Kawase, Y., Makino, K.: Randomized algorithms for online knapsack problems. Theor. Comput. Sci. 562, 395–405 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Han, X., Kawase, Y., Makino, K., Guo, H.: Online removable knapsack problem under convex function. Theor. Comput. Sci. 540, 62–69 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Han, X., Makino, K.: Online removable knapsack with limited cuts. Theor. Comput. Sci. 411(44–46), 3956–3964 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Han, X., Makino, K.: Online minimization knapsack problem. Theor. Comput. Sci. 609, 185–196 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Horiyama, T., Iwama, K., Kawahara, J.: Finite-state online algorithms and their automated competitive analysis. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 71–80. Springer, Heidelberg (2006). doi: 10.1007/11940128_9 CrossRefGoogle Scholar
  8. 8.
    Iwama, K., Taketomi, S.: Removable online knapsack problems. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 293–305. Springer, Heidelberg (2002). doi: 10.1007/3-540-45465-9_26 CrossRefGoogle Scholar
  9. 9.
    Iwama, K., Zhang, G.: Optimal resource augmentations for online knapsack. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX/RANDOM-2007. LNCS, vol. 4627, pp. 180–188. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74208-1_13 CrossRefGoogle Scholar
  10. 10.
    Kawase, Y., Han, X., Makino, K.: Proportional cost buyback problem with weight bounds. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 794–808. Springer, Cham (2015). doi: 10.1007/978-3-319-26626-8_59 CrossRefGoogle Scholar
  11. 11.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    Lueker, G.S.: Average-case analysis of off-line and on-line knapsack problems. In: SODA, pp. 179–188 (1995)Google Scholar
  13. 13.
    Marchetti-Spaccamela, A., Vercellis, C.: Stochastic on-line knapsack problems. Math. Program. 68, 73–104 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Thielen, C., Tiedemann, M., Westphal, S.: The online knapsack problem with incremental capacity. Math. Methods OR 83(2), 207–242 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Software SchoolDalian University of TechnologyDalianChina
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations