Abstract
In early seventies it was shown that the asymptotic approximation ratio of BestFit bin packing is equal to 1.7. We prove that also the absolute approximation ratio for BestFit bin packing is exactly 1.7, improving the previous bound of 1.75. This means that if the optimum needs Opt bins, BestFit always uses at most \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins. Furthermore we show matching lower bounds for all values of Opt, i.e., we give instances on which BestFit uses exactly \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins. Thus we completely settle the worst-case complexity of BestFit bin packing after more than 40 years of its study.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Boyar, J., Dósa, G., Epstein, L.: On the absolute approximation ratio for First Fit and related results. Discrete Appl. Math. 160, 1914–1923 (2012)
Coffman, E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: A survey. In: Hochbaum, D. (ed.) Approximation algorithms. PWS Publishing Company (1997)
Dósa, G., Sgall, J.: First Fit bin packing: A tight analysis. In: Proc. of the 30th Ann. Symp. on Theor. Aspects of Comput. Sci (STACS). LIPIcs, vol. 3, pp. 538–549. Schloss Dagstuhl (2013)
Garey, M.R., Graham, R.L., Johnson, D.S., Yao, A.C.-C.: Resource constrained scheduling as generalized bin packing. J. Combin. Theory Ser. A 21, 257–298 (1976)
Garey, M.R., Graham, R.L., Ullman, J.D.: Worst-case analysis of memory allocation algorithms. In: Proc. 4th Symp. Theory of Computing (STOC), pp. 143–150. ACM (1973)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell System Technical J. 45, 1563–1581 (1966)
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17, 263–269 (1969)
Johnson, D.S.: Near-optimal bin packing algorithms. PhD thesis. MIT, Cambridge, MA (1973)
Johnson, D.S.: Fast algorithms for bin packing. J. Comput. Syst. Sci. 8, 272–314 (1974)
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, 256–278 (1974)
Németh, Z.: A first fit algoritmus abszolút hibájáról (in Hungarian). Eötvös Loránd Univ., Budapest, Hungary (2011)
Sgall, J.: A new analysis of best fit bin packing. In: Kranakis, E., Krizanc, D., Luccio, F. (eds.) FUN 2012. LNCS, vol. 7288, pp. 315–321. Springer, Heidelberg (2012)
Simchi-Levi, D.: New worst case results for the bin-packing problem. Naval Research Logistics 41, 579–585 (1994)
Ullman, J.D.: The performance of a memory allocation algorithm. Technical Report 100, Princeton Univ., Princeton, NJ (1971)
Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press (2011)
Xia, B., Tan, Z.: Tighter bounds of the First Fit algorithm for the bin-packing problem. Discrete Appl. Math. 158, 1668–1675 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dósa, G., Sgall, J. (2014). Optimal Analysis of Best Fit Bin Packing. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_36
Download citation
DOI: https://doi.org/10.1007/978-3-662-43948-7_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-43947-0
Online ISBN: 978-3-662-43948-7
eBook Packages: Computer ScienceComputer Science (R0)