Bin packing problem
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In computational complexity theory, the bin packing problem is a combinatorial NP-hard problem. In it, objects of different volumes must be packed into a finite number of bins of capacity V in a way that minimizes the number of bins used.
There are many variations of this problem, such as 2D packing, linear packing, packing by weight, packing by cost, and so on. They have many applications, such as filling up containers, loading trucks with weight capacity, and creating file backup in removable media.
Since it is NP-hard, the most efficient known algorithms use heuristics to accomplish very good results in most cases, which may not be the optimal solution. The Best Fit Decreasing and First Fit Decreasing strategies use no more than 11/9 OPT + 1 bins (where OPT is the number of bins given by the optimal solution).[1] The simpler of these, the First Fit Decreasing strategy, operates by first sorting the items to be inserted in decreasing order by volume, and then inserting each item into the first bin in the list with sufficient remaining space. The sorting step is relatively expensive, but without it we only achieve the looser bound of 17/10 OPT + 2. A more efficient version of FFD uses no more than 71/60 OPT + 1 bins. [2][3]
Although these simple strategies are often good enough, efficient approximation algorithms have been demonstrated that can solve the bin packing problem within any fixed percentage of the optimal solution for sufficiently large inputs (this is called an asymptotic polynomial-time approximation scheme). This is an advantage the problem has over many other common NP-hard problems, some of which cannot be approximated within any constant factor at all.
[edit] See also
[edit] References
- ^ M. Yue, "A simple proof of the inequality FFD(L) ≤ (11/9)OPT(L) + 1, for all L, for the FFD bin-packing algorithm", Acta Mathematicae Applicatae Sinica 7 (1991), pp. 321–331.
- ^ Michael R. Garey and David S. Johnson, "A 71/60 theorem for bin packing", Journal of Complexity, Vol. 1 (1985), pp. 65–106.
- ^ Yue Minyi and Zhang Lei, "A simple proof of the inequality MFFD(L) ≤ 71/60 OPT(L) + 1, L for the MFFD bin-packing algorithm", Acta Mathematicae Applicatae Sinica 11 (1995), pp. 318–330.
- Michael R. Garey and David S. Johnson (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A4.1: SR1, p. 226.
- Vijay V. Vazirani, Approximation Algorithms. Springer. ISBN 3-540-65367-8. p. 74.
- David S. Johnson, Alan J. Demers, Jeffrey D. Ullman, M. R. Garey, Ronald L. Graham. Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. SICOMP, Volume 3, Issue 4. 1974.
