Branch and bound
Imagine you have a bunch of toys that you want to organize in your toy chest. You know the toys come in different shapes and sizes, and you want to put them in the chest in such a way that they take up the least amount of space possible.
However, you don't know where to start, so you decide to use a strategy called "branch and bound." This strategy involves breaking up the problem into smaller sub-problems (i.e., "branches"), solving them one by one, and then tying them all back together (i.e., "bounding").
So, you start by taking one toy out of the pile and placing it in the chest. This gives you a starting point, but there are still many ways that the rest of the toys can be arranged around it. For example, you could put a small toy next to it or a large toy on top of it. The possibilities are endless.
To solve this problem, you need to create "branches" that represent all of the different possibilities for how the toys can be arranged around the first toy. In other words, you need to create a bunch of different "sub-problems" that each have the first toy already placed, but with different combinations of the other toys.
Once you have created all of the different branches, you can start solving each sub-problem one by one. This involves putting more toys in the chest and trying out different arrangements until you find the best one.
As you go through each sub-problem, you keep track of the best solution you have found so far, and use that to "bound" the problem. This means that you eliminate any sub-problems that could not possibly lead to a better solution than the one you have already found.
For example, if you have found a solution that takes up 50% of the space in the chest, any sub-problems that would take up more than 50% of the space can be eliminated because they are not better than what you already have.
Finally, when you have solved all of the sub-problems, you tie them all back together by selecting the best solution from each branch and comparing them to find the one that takes up the least amount of space.
In summary, branch and bound is a clever technique for solving complex problems by breaking them down into smaller sub-problems, solving each one individually, and then combining them back together to find the best solution. It's like putting toys in a toy chest, but with a lot more math involved.
However, you don't know where to start, so you decide to use a strategy called "branch and bound." This strategy involves breaking up the problem into smaller sub-problems (i.e., "branches"), solving them one by one, and then tying them all back together (i.e., "bounding").
So, you start by taking one toy out of the pile and placing it in the chest. This gives you a starting point, but there are still many ways that the rest of the toys can be arranged around it. For example, you could put a small toy next to it or a large toy on top of it. The possibilities are endless.
To solve this problem, you need to create "branches" that represent all of the different possibilities for how the toys can be arranged around the first toy. In other words, you need to create a bunch of different "sub-problems" that each have the first toy already placed, but with different combinations of the other toys.
Once you have created all of the different branches, you can start solving each sub-problem one by one. This involves putting more toys in the chest and trying out different arrangements until you find the best one.
As you go through each sub-problem, you keep track of the best solution you have found so far, and use that to "bound" the problem. This means that you eliminate any sub-problems that could not possibly lead to a better solution than the one you have already found.
For example, if you have found a solution that takes up 50% of the space in the chest, any sub-problems that would take up more than 50% of the space can be eliminated because they are not better than what you already have.
Finally, when you have solved all of the sub-problems, you tie them all back together by selecting the best solution from each branch and comparing them to find the one that takes up the least amount of space.
In summary, branch and bound is a clever technique for solving complex problems by breaking them down into smaller sub-problems, solving each one individually, and then combining them back together to find the best solution. It's like putting toys in a toy chest, but with a lot more math involved.
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