4.3.2 Fully polynomial time approximation scheme.4.1 Dynamic programming in-advance algorithm.The name 'knapsack problem' dates back to the early works of the mathematician Tobias Dantzig (1884–1956), and refers to the commonplace problem of packing the most valuable or useful items without overloading the luggage. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. (Solution: if any number of each box is available, then three yellow boxes and three grey boxes if only the shown boxes are available, then all except for the green box.) Example of a one-dimensional (constraint) knapsack problem: which boxes should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15 kg? A multiple constrained problem could consider both the weight and volume of the boxes.

Small knapsack