Bin Packing Approximation

In this video, we discuss the Bin Packing problem. We show that Bin Packing allows for a 1.5-approximation. We also present a matching lower bound. So both upper and lower bounds are optimal, unless P is equal to NP.

[@gemini_537]

Gemini 1.5 Pro: This video is about the Bin Packing Problem and how to solve it with an approximation algorithm.
The video starts with a scenario where you are running a drone transportation company. The drone has a carrying capacity of one kilogram, and you want to pack the items into boxes as tightly as possible to save on the number of journeys. The items include an apple weighing 500 grams, an orange weighing 300 grams, 400 grams of strawberries, and 700 grams of grapes.
The video then introduces a simple algorithm called First Fit. This algorithm considers the items in whatever order and places them into bins one by one. The goal is to place each item in the leftmost bin that it fits into. For example, with the items listed above, the first item would go into the first bin, the second item would not fit into the first bin so it goes into the second bin, and so on. This algorithm uses five bins in total.
The video then analyses the First Fit algorithm and proves that it is a two approximation algorithm for bin packing. This means that the algorithm uses at most two times as many bins as the optimal solution. The video also discusses an improvement to the First Fit algorithm by sorting the elements in decreasing order. This improvement can achieve an approximation ratio of three over two.
The video concludes by proving that it is NP-hard to approximate bin packing to any constant less than 1.5. This means that there is no polynomial time algorithm that can guarantee an approximation ratio better than 1.5. The video also mentions that First Fit with sorting is a 1.5-approximation algorithm.

[@dvir-ross]

Great video! Thanks!
You definitly deserve more recognition.

[@maherjarrah2758]

Thank you! waiting for a 3D Bin Packing video :)

[@USMANHAMEED-sy1gx]

Respect 🙏

[@USMANHAMEED-sy1gx]

Can you please also post 2D Bin Packing?

[@loganwenzel1615]

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