Analysis of Approximate Algorithms
The best fit decreasing and first fit decreasing strategies are among the simplest heuristic algorithms for solving the bin packing problem. They have been shown to use no more than 11/9 OPT + 1 bins (where OPT is the number of bins given by the optimal solution). The simpler of these, the First Fit Decreasing (FFD) strategy, operates by first sorting the items to be inserted in decreasing order by their sizes, and then inserting each item into the first bin in the list with sufficient remaining space. Without the sorting step, we only achieve the looser bound of 17/10 OPT + 2. Sometimes, however, one does not have the option to sort the input, for example, when faced with an online bin packing problem. In 2007, it was proven that the bound 11/9 OPT + 6/9 for FFD is tight. MFFD (a variant of FFD) uses no more than 71/60 OPT + 1 bins (i.e. bounded by about 1.18×opt, compared to about 1.22×opt for FFD). In 2010, an upper bound for the asymptotic performance ratio was decreased to 17/10 OPT + 7/10 for FF and for the absolute performance ratio - to 12/7 OPT.
It is NP-hard to distinguish whether OPT is 2 or 3, thus for all ε > 0, bin packing is hard to approximate within 3/2 − ε. (If such an approximation exists, one could determine whether n non-negative integers can be partitioned into two sets with the same sum in polynomial time. However, this problem is known to be NP-hard.) Consequently, the bin packing problem does not have a polynomial-time approximation scheme (PTAS) unless P = NP. On the other hand, for any 0 < ε ≤ 1, it is possible to find a solution using at most (1 + ε)OPT + 1 bins in polynomial time. This approximation type is known as asymptotic PTAS.
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