## Saturday 19 July 2014

### Discrete Geometry - Bin Packing Problem

This post is a little irrelevant to general contents of my blog, but I found this to be a interesting geometry problem and it does have some ties with Computational Geometry, which is form a of theoretical computer science. There is additionally some connection with Computational Complexity Theory too. The Bin Packing Problem isn't difficult to explain, and yet can be difficult to find a optimal solution.

With Discrete Mathematics, I personally find that the branches within this field are more accessible but the problems are difficult enough to be interesting and form a field of serious mathematical study. I'm only a amateur mathematician and a student, so if there are any problems then please highlight them in the comments section.

Bin Packing Problem:

The Bin Packing Problem is an example of a optimization problem which has a surprisingly large number of applications, especially in logistics and data management. The Bin Packing Problem asks to minimize the number of bins needed to pack a certain number and given volume for a list of objects. These objects can vary in size, but the bin volumes will remain fixed. There are some programs which will give valid suggestions for the most optimal method, however, the problem is a NP- Hard Combinatorial class type.

The sum of the sizes of the items must be less than or equal to the total volume of the bins being used. The size of the items can never be greater than the total volume of the bins. If the volume of one bin is reached, then a new bin will need to be used. The problem looks to find a packing method which will reduce the number of bins needed to provide a optimal method.

The First-Fit Algorithm is the best algorithm which has been used for the bin packing problem. The First-Fit Algorithm is an example of a greedy approximation algorithm, in that the items will processed in any given order. The algorithm will place as many items as possible into the first bin, and then create a new bin if no other additional bins can be found. The process is then repeated for the rest of the items.

The First-Fit Algorithm has a approximation factor of 2 (APX). The approximation factor is used for NP-Hard problems, since it is very unlikely that a efficient algorithm will be produced to solve the problem directly, and therefore a P class algorithm can be developed in order to find a approximate answer. The approximation factor of 2, means that the algorithm will never use more than twice the number of least bins needed for the bin packing problem. For instance, if the number of bins needed was 2, then the algorithm will never use more than 4.

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