# Algorithms – Is this problem NP-Complete (filling the trash with separable elements and penalty)?

The problem looks a bit like the trash. I will describe it with a similar name:

• you have $$B$$ trash bins of the same size, $$V$$, or $$V$$ is a positive integer

• This problem has elements, but also "pieces" of elements: each article is considered a piece and each piece must always have a "valid size": the size must be a positive integer.

• You have a set, $$S_1$$, items that must all be placed in bins, but can be separated into "pieces" of any valid size (that is, the sum of the sizes of the pieces of an article must always be equal to the original size of the article)

• You have a second set, $$S_2$$, with non-separable elements of the same size, the cardinality of this set is sufficiently large / unlimited

• If you place a piece of S_1 in a basket, you must also place exactly one item of $$S_2$$ in the same bin

In one case, is it possible to place all the elements of $$S_1$$ in the bins without exceeding the capacity of the bins?

More simply: is bin-packing always NP-Complete if you allow the separation of the pieces into pieces, but there is a "penalty" to have more pieces in each bin, where you are obliged to place a "shutter" between each piece in a trash?

I've been considering trying to reduce packing, scheduling, partitioning to 3 partitions, 3 columns, SAT, TSP, but I do not see how the make. Also, trying to solve the problem in the allotted time. I can only think of approximation algorithms such as greedily placing the largest item in the bin with the largest remaining capacity.

Any response or observation on this subject would be very appreciated.