Abstract
We consider a memory allocation problem. This problem can be modeled as a version of bin packing where items may be split, but each bin may contain at most two (parts of) items. This problem was recently introduced by Chung et al. (Theory Comput. Syst. 39(6):829–849, 2006). We give a simple \(\frac{3}{2}\) -approximation algorithm for this problem which is in fact an online algorithm. This algorithm also has good performance for the more general case where each bin may contain at most k parts of items. We show that this general case is strongly NP-hard for any k≥3. Additionally, we design an efficient approximation algorithm, for which the approximation ratio can be made arbitrarily close to \(\frac{7}{5}\) .
Article PDF
References
Babel, L., Chen, B., Kellerer, H., Kotov, V.: Algorithms for on-line bin-packing problems with cardinality constraints. Discrete Appl. Math. 143(1–3), 238–251 (2004)
Caprara, A., Kellerer, H., Pferschy, U.: Approximation schemes for ordered vector packing problems. Nav. Res. Logist. 92, 58–69 (2003)
Chung, F., Graham, R., Mao, J., Varghese, G.: Parallelism versus memory allocation in pipelined router forwarding engines. Theory Comput. Syst. 39(6), 829–849 (2006)
Epstein, L.: Online bin packing with cardinality constraints. SIAM J. Discrete Math. 20(4), 1015–1030 (2006)
Epstein, L., Levin, A.: AFPTAS results for common variants of bin packing: A new method to handle the small items. Manuscript (2007)
Epstein, L., van Stee, R.: Approximation schemes for packing splittable items with cardinality constraints. In: Fifth Workshop on Approximation and Online Algorithms (WAOA 2007). Lecture Notes in Computer Science, vol. 4927, pp. 232–245. Springer, Berlin (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Johnson, D.S.: Fast algorithms for bin packing. J. Comput. Syst. Sci. 8(3), 272–314 (1974)
Kellerer, H., Pferschy, U.: Cardinality constrained bin-packing problems. Ann. Oper. Res. 92, 335–348 (1999)
Krause, K.L., Shen, V.Y., Schwetman, H.D.: Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. J. ACM 22(4), 522–550 (1975)
Krause, K.L., Shen, V.Y., Schwetman, H.D.: Errata: “Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems”. J. ACM 24(3), 527 (1977)
Mao, J., Graham, R.L.: Parallel resource allocation of splittable items with cardinality constraints. Manuscript
Shachnai, H., Yehezkely, O.: Fast asymptotic FPTAS for packing fragmentable items with costs. In: Proc. of the 16th International Symposium on Fundamentals of Computation Theory (FCT2007), pp. 482–493 (2007)
Shachnai, H., Tamir, T., Yehezkely, O.: Approximation schemes for packing with item fragmentation. Theory Comput. Syst. 43(1), 81–98 (2008)
van Vliet, A.: An improved lower bound for online bin packing algorithms. Inf. Process. Lett. 43(5), 277–284 (1992)
Yao, A.C.C.: New algorithms for bin packing. J. ACM 27, 207–227 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper appeared in the Proceedings of Tenth Workshop on Algorithms and Data Structures (WADS 2007). LNCS, vol. 4619, pp. 362–373. Springer, 2007.
Work performed while R. van Stee was at University of Karlsruhe, Germany. Research supported by Alexander von Humboldt Foundation.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Epstein, L., van Stee, R. Improved Results for a Memory Allocation Problem. Theory Comput Syst 48, 79–92 (2011). https://doi.org/10.1007/s00224-009-9226-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-009-9226-2