So here we're going to also take this chance to give you some ideas, some more deeper understanding about the knapsack problem. Second is that knapsack is just so important so it deserves a lot of attentions. For each sub problem if there is a unique feature then don't use the simplest medicine use something else that is actually faster. First, we want to give you an illustration about the idea I just mentioned. So here we're going to give you another example is that we are going to use the branch-and-bound to solve the knapsack problem. Just use your observation then it may be good enough. So one example that you already observed is that sometimes when you have only two variables, then in that case for solving each linear program, you don't really need to do simplex method. When it is possible, don't use the simplex method. You should try to speed up the process for solving each linear program. But if your problem, if your sub problems they have some unique features that you don't need to use the simplex method, then you should do that. So the simplex method is always valid for solving a linear program. And it happens that sometimes we don't need to use the general simplex method. But there's one thing that I would like to remind you is that when we are doing branch-and-bound for each subproblem we are solving a linear program pretty much, okay? And for such a linear program, in many cases, we somehow need to solve that particular optimization problem, we somehow need to find a solution. So that was a general description about branch-and-bound.
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