An Improved Algorithm for Optimal Bin Packing

Given a set of numbers, and a set of bins of fixed capacity, the NP-complete problem of bin packing is to find the minimum number of bins needed to contain the numbers, such that the sum of the numbers assigned to each bin does not exceed the bin capacity. We present two improvements to our previous bin-completion algorithm. The first speeds up the constant factor per node generation, and the second prunes redundant parts of the search tree. The resulting algorithm appears to be asymptotically faster than our original algorithm. On problems with 90 elements, it runs over 14 times faster. Furthermore, the ratios of node generations and running times both increase with increasing problem size. 1 Introduction and Overview Given a set of numbers, and a fixed bin capacity, the bin-packing problem is to assign each number to a bin so that the sum of the numbers assigned to each bin does not exceed the bin capacity. An optimal solution uses the fewest number of bins. For example, given the to another, for a total of two bins. This is an optimal solution to this instance, since the sum of all the numbers, 198, is greater than 100, and hence at least two bins are required. An example application is given a set of orders for wire of varying lengths, and a standard length in which it is manufactured, how to cut up the minimum number of standard lengths to fill the orders. Bin packing was one of the earliest problems shown to be NP-complete[Garey & Johnson, 1979]. The vast majority of the literature on this problem concerns polynomial-time approximation algorithms, such as first-fit decreasing (FFD) and best-fit decreasing (BFD), and the quality of the solutions they compute. First-fit decreasing sorts the numbers in decreasing order, orders the bins, and assigns each number in turn to the first bin in which it fits. Best-fit decreasing sorts the numbers in decreasing order and then assigns each number in turn to the fullest bin in which it fits. First-fit decreasing requires three bins to pack the set of numbers above, while best-fit decreasing packs them into two bins. Both algorithms run in O(nlogn) time. In this paper we are concerned with finding optimal solutions, for several reasons. In applications with small numbers of bins, even one extra bin is relatively expensive. In addition, being able to find optimal solutions to problem …