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First-fit-decreasing (FFD) is an algorithm for bin packing.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.
Taking R=1/6 yields that the weight of all FF bins is at least 2/3. On the other hand, the weight of every bin in the optimal packing is at most 1+R = 7/6, since each such bin has at most one item larger than 1/2. Therefore, the total weight of all items is at most 7/6*OPT, and the number of FF bins is at most 2+(7/6*OPT/(2/3)) = 7/4*OPT+2.
Bin-packing with fragmentation or fragmentable object bin-packing is a variant of the bin packing problem in which it is allowed to break items into parts and put each part separately on a different bin. Breaking items into parts may allow for improving the overall performance, for example, minimizing the number of total bin.
People are given n unit squares and have to pack them into the smallest possible container, where the container type varies: Packing squares in a square: Optimal solutions have been proven for n from 1-10, 14-16, 22-25, 33-36, 62-64, 79-81, 98-100, and any square integer. The wasted space is asymptotically O(a 3/5).
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Next-k-Fit is a variant of Next-Fit, but instead of keeping only one bin open, the algorithm keeps the last bins open and chooses the first bin in which the item fits. For k ≥ 2 {\displaystyle k\geq 2} , NkF delivers results that are improved compared to the results of NF, however, increasing k {\displaystyle k} to constant values larger than ...