11 Facts About Knapsack problem

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.

The Knapsack 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.

Knapsack problem has been studied for more than a century, with early works dating as far back as 1897.

Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity.

Unbounded knapsack problem places no upper bound on the number of copies of each kind of item and can be formulated as above except for that the only restriction on is that it is a non-negative integer.

One example of the unbounded knapsack problem is given using the figure shown at the beginning of this article and the text "if any number of each box is available" in the caption of that figure.

Knapsack problem is interesting from the perspective of computer science for many reasons:.

One theme in research literature is to identify what the "hard" instances of the knapsack problem look like, or viewed another way, to identify what properties of instances in practice might make them more amenable than their worst-case NP-complete behaviour suggests.

Fully polynomial time approximation scheme for the knapsack problem takes advantage of the fact that the reason the problem has no known polynomial time solutions is because the profits associated with the items are not restricted.

The Knapsack problem was introduced by Gallo, Hammer, and Simeone in 1980, however the first treatment of the Knapsack problem dates back to Witzgall in 1975.

Generalization of subset sum Knapsack problem is called multiple subset-sum Knapsack problem, in which multiple bins exist with the same capacity.