Abstract
In this paper, we study a bin packing problem in which the sizes of items are determined by k linear constraints, and the goal is to decide the sizes of items and pack them into a minimal number of unit sized bins. We first provide two scenarios that motivate this research. We show that this problem is NP-hard in general, and propose several algorithms in terms of the number of constraints. If the number of constraints is arbitrary, we propose a 2-approximation algorithm, which is based on the analysis of the Next Fit algorithm for the bin packing problem. If the number of linear constraints is a fixed constant, then we obtain a PTAS by combining linear programming and enumeration techniques, and show that it is an optimal algorithm in polynomial time when the number of constraints is one or two. It is well known that the bin packing problem is strongly NP-hard and cannot be approximated within a factor 3 / 2 unless P = NP. This result implies that the bin packing problem with a fixed number of constraints may be simper than the original bin packing problem. Finally, we discuss the case when the sizes of items are bounded.
Similar content being viewed by others
References
Coffman EG Jr, Csirik J, Galambos G, Martello S, Vigo D (2013) Bin packing approximation algorithms: survey and classification. In: Pardalos PM, Du D-Z, Graham RL (eds) Handbook of combinatorial optimization. Springer, New York, pp 455–531
Coffman EG Jr, Csirik J, Leung JY-T (2006) Variable-sized bin packing and bin covering. In: Gonzales T (ed) Handbook of approximation algorithms and metaheuristics. Taylor and Francis Books, Boca Raton, pp 34-1–34-11 (chapter 34)
Coffman EG Jr, Garey MR, Johnson DS (1997) Approximation algorithms for bin packing: a survey. In: Hochbaum DS (ed) Approximation algorithms for np-hard problems. PWS, Boston, pp 46–93
Correa JR, Epstein L (2008) Bin packing with controllable item sizes. Inf Comput 206(8):1003–1016
Dantzig GB (1951) Maximization of a linear function of variables subject to linear inequalities. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, London, pp 339–347
Dósa G, Sgall J (2013) First fit bin packing: a tight analysis. In 30th symposium on theoretical aspects of computer science (STACS), pp 538–549
Dósa G, Li R, Han X, Tuza Z (2013) Tight absolute bound for first fit decreasing bin-packing: \(FFD(I)\le 11/9OPT(I) + 6/9\). Theor Comput Sci 510:13–61
Fernandez de la Vega W, Lueker GS (1981) Bin packing can be solved within 1 + \(\epsilon \) in linear time. Combinatorica 1(4):349–355
Friesen DK, Langston MA (1986) Variable sized bin packing SIAM. J Comput 15(1):222–230
Garey MR, Graham RL, Ullman JD (1972) Worst-case analysis of memory allocation algorithms. Proceedings of the 4th annual ACM symposium on theory of computing STOC ’72. Plenum Press, New York, pp 143–150.
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of np-completeness. W. H. Freeman and Co., San Francisco
Hochbaum DS (1996) Approximation algorithms for NP-hard problems. PWS, Boston
Johnson D S (1973) Near-optimal bin packing algorithms. Ph.D. thesis, MIT, Cambridge, MA
Johnson DS, Demers A, Ullman JD, Garey MR, Graham RL (1974) Worst-case performance bounds for simple one-dimensional packing algorithms SIAM. J Comput 3(4):299–325
Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395
Khachiyan LG (1980) Polynomial algorithms in linear programming USSR. Comput Math Math Phys 20(1):53–72
Korte B, Vygen J (2012) Combinatorial optimization: theory and algorithms, 4th edn. Springer, Berlin
Nip K, Wang Z, Wang Z (2016) Scheduling under linear constraints. Eur J Oper Res 253(2):290–297
Rhee WT (1988) Optimal bin packing with items of random sizes. Math Oper Res 13(1):140–151
Rhee WT, Talagrand M (1989) Optimal bin packing with items of random sizes II SIAM. J Comput 18(1):139–151
Ullman JD (1971) The performance of a memory allocation algorithm. Technical report, Princeton University, Princeton
Ye Y (1997) Interior point algorithms: theory and analysis. Wiley, London
Acknowledgements
We thank two anonymous reviewers for their constructive comments. Zhenbo Wang’s research has been supported by NSFC No. 11371216.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, Z., Nip, K. Bin packing under linear constraints. J Comb Optim 34, 1198–1209 (2017). https://doi.org/10.1007/s10878-017-0140-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-017-0140-2