The standard method to solve an integer programming is called Branch-and-Bound. Where I\') return solver.Objective().Value() Suppose the specs of the items are given in the following table: However, this does not guarantee an optimal solution to the 0–1 knapsack problem, as demonstrated by the following counter example. This algorithm is a greedy algorithm, and is actually the solution to the fractional knapsack problem. A natural idea is to calculate a utility/weight ratio for each item, then try to fit the items that has higher utility score per unit weight until hitting the weight threshold. The main tradeoff of the knapsack problem is between getting as much utility score as possible while satisfying the constrained weight limit. In real life, the 0–1 knapsack problem can be seen as deciding if a flashlight should be selected, whereas the fractional problem might be considering how much of the 1 gallon of water should be selected. This is a good distinction between the 0–1 knapsack problem where each item is a whole, and the fractional knapsack problem where a fractional part of an item can be selected. One caveat is that each item can be either selected or not selected, but it can not be half selected or fractionally selected. You would like to find the subset of items, the total weight of which is below a threshold W and the total utility score of which is maximized.įrom the description, the problem sounds simple enough. Each item has a weight and an utility score. Suppose you are going on a trip, and you have a list of n items you would like to carry with you in your backpack (or knapsack, if you will). Let’s recap what the knapsack problem is. The more reason we want to start with it! What’s more, knapsack problem is in fact a NP-hard problem! However, it is one of the NP hard problem we can solve pretty efficiently. As an optimization person, knapsack problem is one of the first problems you learn in integer programming class. But even before Leetcode, knapsack was covered in the introduction of integer programming classes and perhaps higher level computer science classes, due to its recursive nature and easy problem setup. Knapsack problem is perhaps widely-known as one of the medium level Leetcode problem. In this article, we are going to dive deeper into the difference between dynamic programming and integer programming with the interesting and well-studied problem of knapsack problem. Often times, one problem could be solved with multiple different approaches, and this is where different underlying solution philosophies meet each other, which I find very interesting. You can classify the field base on many different criteria: Linear vs nonlinear, convex vs non-convex, continuous vs discrete, etc. The field of optimization encompasses many different fields of models and algorithms.
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