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The bin packing problem - a dual problem in which the total sum in each subset is bounded, but k is flexible; the goal is to find a partition with the smallest possible k. The optimization objectives are closely related: the optimal number of d-sized bins is at most k, iff the optimal size of a largest subset in a k-partition is at most d. [9]
The values (), …, of the partition function (1, 2, 3, 5, 7, 11, 15, and 22) can be determined by counting the Young diagrams for the partitions of the numbers from 1 to 8. In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n.
LDM always returns a partition in which the largest sum is at most 7/6 times the optimum. [4] This is tight when there are 5 or more items. [2] On random instances, this approximate algorithm performs much better than greedy number partitioning. However, it is still bad for instances where the numbers are exponential in the size of the set. [5]
The partition problem is NP hard. This can be proved by reduction from the subset sum problem. [6] An instance of SubsetSum consists of a set S of positive integers and a target sum T; the goal is to decide if there is a subset of S with sum exactly T.
The limit on partition size was dictated by the 8-bit signed count of sectors per cluster, which originally had a maximum power-of-two value of 64. With the standard hard disk sector size of 512 bytes, this gives a maximum of 32 KB cluster size, thereby fixing the "definitive" limit for the FAT16 partition size at 2 GB for sector size 512.
The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part.
The exact MILP results for 3,4,5,6,7 correspond to the lower bound. For k >7, no exact results are known, but the difference between the lower and upper bound is less than 0.3%. When the parameter is the number of subsets ( m ), the approximation ratio is exactly 2 − 1 m {\displaystyle 2-{\frac {1}{m}}} .
Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem.