Skip to main content

Approximating Robust Bin Packing with Budgeted Uncertainty

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11646)


We consider robust variants of the bin-packing problem where the sizes of the items can take any value in a given uncertainty set \(U\subseteq \times _{i=1}^n[\overline{a}_i,\overline{a}_i+\hat{a}_i]\), where \(\overline{a}\in [0,1]^n\) represents the nominal sizes of the items and \(\hat{a}\in [0,1]^n\) their possible deviations. We consider more specifically two uncertainty sets previously studied in the literature. The first set, denoted \(U^\varGamma \), contains scenarios in which at most \(\varGamma \in \mathbb {N}\) items deviate, each of them reaching its peak value \(\overline{a}_i+\hat{a}_i\), while each other item has its nominal value \(\overline{a}_i\). The second set, denoted \(U^\varOmega \), bounds by \(\varOmega \in [0,1]\) the total amount of deviation in each scenario. We show that a variant of the next-fit algorithm provides a 2-approximation for model \(U^\varOmega \), and a \(2(\varGamma +1)\) approximation for model \(U^\varGamma \) (which can be improved to 2 approximation for \(\varGamma =1\)). This motivates the question of the existence of a constant ratio approximation algorithm for the \(U^\varGamma \) model. Our main result is to answer positively to this question by providing a 4.5 approximation for \(U^\varGamma \) model based on dynamic programming.


  • Bin-packing
  • Robust optimization
  • Approximation algorithm
  • Next-fit
  • Dynamic programming

This research has benefited from the support of the ANR project ROBUST [ANR-16-CE40-0018].

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-24766-9_6
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   79.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-24766-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   99.99
Price excludes VAT (USA)


  1. 1.

    \(U^\varGamma \) is often defined alternatively in the literature, as the polytope \(\{a\in \times _{i\in [n]} [\bar{a}_i, \bar{a}_i + \hat{a}_i]\mid \sum _{i\in [n]}(a_i -\bar{a}_i)/\hat{a}_i \le \varGamma \}\). For the bin-packing problem, one readily verifies using classical arguments that the two definitions lead to the same optimization problem.

  2. 2.

    In general, if we have a polynomial time additive approximation algorithm using \(OPT + f(OPT)\) bins and polynomial time \(\rho \)-approximation algorithm for \(\varGamma \textsc {RBP}\) with small values then our algorithm uses \(OPT(\rho + 1) + f(OPT)\) bins for \(\varGamma \textsc {RBP}\) in polynomial time.


  1. Aissi, H., Bazgan, C., Vanderpooten, D.: Min-max and min-max regret versions of combinatorial optimization problems: a survey. Eur. J. Oper. Res. 197(2), 427–438 (2009)

    CrossRef  MathSciNet  Google Scholar 

  2. Álvarez-Miranda, E., Ljubic, I., Toth, P.: A note on the Bertsimas & Sim algorithm for robust combinatorial optimization problems. 4OR 11(4), 349–360 (2013).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Balogh, J., Békési, J., Dósa, G., Sgall, J., van Stee, R.: The optimal absolute ratio for online bin packing. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (2015)

    Google Scholar 

  4. Balogh, J., Békési, J., Dósa, G., Epstein, L., Levin, A.: A new and improved algorithm for online bin packing. In: Azar, Y., Bast, H., Herman, G. (eds.) ESA. LIPIcs, vol. 112, pp. 5:1–5:14. Dagstuhl, Germany (2018)

    Google Scholar 

  5. Basu Roy, A., Bougeret, M., Goldberg, N., Poss, M.: Approximating the robust bin-packing with budget uncertainty (2019).

  6. Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98(1–3), 49–71 (2003)

    CrossRef  MathSciNet  Google Scholar 

  7. Bougeret, M., Pessoa, A.A., Poss, M.: Robust scheduling with budgeted uncertainty. Discrete Appl. Math. 261(31), 93–107 (2019)

    CrossRef  MathSciNet  Google Scholar 

  8. Dexter, F., Macario, A., Traub, R.D.: Which algorithm for scheduling add-on elective cases maximizes operating room utilization? Use of bin packing algorithms and fuzzy constraints in operating room management. Anesthesiology 91, 1491–1500 (1999)

    CrossRef  Google Scholar 

  9. Goetzmann, K.-S., Stiller, S., Telha, C.: Optimization over integers with robustness in cost and few constraints. In: Solis-Oba, R., Persiano, G. (eds.) WAOA 2011. LNCS, vol. 7164, pp. 89–101. Springer, Heidelberg (2012).

    CrossRef  MATH  Google Scholar 

  10. Gounaris, C.E., Wiesemann, W., Floudas, C.A.: The robust capacitated vehicle routing problem under demand uncertainty. Oper. Res. 61(3), 677–693 (2013)

    CrossRef  MathSciNet  Google Scholar 

  11. Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J. Comput. 3(4), 299–325 (1974)

    CrossRef  MathSciNet  Google Scholar 

  12. Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (SFCS 1982), pp. 312–320, November 1982

    Google Scholar 

  13. Kasperski, A., Zieliński, P.: On the approximability of minmax (regret) network optimization problems. Inform. Process. Lett. 109(5), 262–266 (2009)

    CrossRef  MathSciNet  Google Scholar 

  14. Kasperski, A., Zielinski, P.: On the approximability of robust spanning tree problems. Theor. Comput. Sci. 412(4–5), 365–374 (2011).

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Poss, M.: Robust combinatorial optimization with knapsack uncertainty. Discrete Optim. 27, 88–102 (2018).

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Simchi-Levi, D.: New worst-case results for the bin-packing problem. Naval Res. Logist. 41, 579–585 (1994)

    CrossRef  MathSciNet  Google Scholar 

  17. Song, G., Kowalczyk, D., Leus, R.: The robust machine availability problem–bin packing under uncertainty. IISE Trans. 50(11), 997–1012 (2018).

    CrossRef  Google Scholar 

  18. Tadayon, B., Smith, J.C.: Algorithms and complexity analysis for robust single-machine scheduling problems. J. Sched. 18(6), 575–592 (2015)

    CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Michael Poss .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Basu Roy, A., Bougeret, M., Goldberg, N., Poss, M. (2019). Approximating Robust Bin Packing with Budgeted Uncertainty. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-24765-2

  • Online ISBN: 978-3-030-24766-9

  • eBook Packages: Computer ScienceComputer Science (R0)