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Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 350–364 | Cite as

Tight bounds for NF-based bounded-space online bin packing algorithms

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Abstract

In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1 / 2. Using an NF-based online algorithm the authors proved an ACR of \(13/9 = 1.44\ldots \) for any given buffer size not less than 1. They also gave a lower bound of \(4/3=1.33\ldots \) for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243,  and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of \(h_\infty (r)\) for the r-parametric problem when the buffer capacity is 3. Since \(h_\infty (2) = 1.42312\ldots \), our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r.

Keywords

Semi-online bin-packing Bounded-space Asymptotic competitive ratio Next-fit Lower bound 

Notes

Acknowledgements

The authors would like to thank an anonymous referee for the useful comments on a previous version of this paper. Her/his comments have helped us to improve strongly the presentation of the paper.

References

  1. Balogh J, Békési J, Galambos G (2012) New lower bounds for certain classes of bin packing algorithms. Theor Comput Sci 440–441:1–13MathSciNetCrossRefMATHGoogle Scholar
  2. Balogh J, Békési J, Galambos G, Reinelt G (2014) Online bin packing with restricted repacking. J Comb Optim 27(1):115–131MathSciNetCrossRefMATHGoogle Scholar
  3. Epstein L, Kleiman E (2009) Resource augmented semi-online bounded space bin packing. Discrete Appl Math 157:2785–2798MathSciNetCrossRefMATHGoogle Scholar
  4. Galambos G (1986) Parametric lower bound for online bin packing. SIAM J Algebr Discrete Methods 7:362–367CrossRefMATHGoogle Scholar
  5. Galambos G, Woeginger GJ (1993) Repacking helps in bounded space online bin packing. Computing 49:329–338MathSciNetCrossRefMATHGoogle Scholar
  6. Grove EF (1995) Online bin packing with lookahead. In: Proceedings of the sixth annual ACM-SIAM Symposium on Discrete Algorithms, pp 430–436Google Scholar
  7. Gutin G, Jensen T, Yeo A (2005) Batched bin packing. Discrete Optim 2(1):71–82MathSciNetCrossRefMATHGoogle Scholar
  8. Heydrich S, van Stee R (2016) Beating the harmonic lower bound for online bin packing. In: Rabani Y, Chatzigiannakis I, Mitzenmacher M, Sangiorgi D (eds) ICALP 2016, Leibniz international proceedings in informatics, vol 55, pp 41:1–41:14Google Scholar
  9. Johnson DS (1973) Near-optimal bin-packing algorithms. Doctoral Thesis, MIT, CambridgeGoogle Scholar
  10. Seiden S (2002) On the online bin packing problem. J ACM 49:640–671MathSciNetCrossRefMATHGoogle Scholar
  11. Sylvester J (1880) On a point in the theory of vulgar fractions. Am J Math 3:332–335MathSciNetCrossRefMATHGoogle Scholar
  12. van Vliet A (1992) An improved lower bound for online bin packing algorithms. Inf Proc Lett 43:274–284MATHGoogle Scholar
  13. Zhang M, Han X, Lan Y, Ting H-F (2017) Online bin packing problem with buffer and bounded size revisited. J Comb Optim 33(2):530–542MathSciNetCrossRefMATHGoogle Scholar
  14. Zheng F, Huo L, Zhang E (2015) NF-based algorithms for online bin packing with buffer and bounded item size. J Comb Optim 30(2):360–369MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Applied Informatics, Gyula Juhász Faculty of EducationUniversity of SzegedSzegedHungary

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