Knapsack problem

The knapsack problem or rucksack 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. 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 quadratic knapsack problem (QKP) maximizes a quadratic objective function subject to a binary and linear capacity constraint. The knapsack problem or rucksack 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. 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 problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography, applied mathematics, and daily fantasy sports. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The name 'knapsack problem' dates back to the early works of mathematician Tobias Dantzig (1884–1956), and refers to the commonplace problem of packing the most valuable or useful items without overloading the luggage. A 1998 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem. Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, selection of investments and portfolios, selection of assets for asset-backed securitization, and generating keys for the Merkle–Hellman and other knapsack cryptosystems. One early application of knapsack algorithms was in the construction and scoring of tests in which the test-takers have a choice as to which questions they answer. For small examples, it is a fairly simple process to provide the test-takers with such a choice. For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve a maximum possible score of 100 points. However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices. Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points. The students are asked to answer all of the questions to the best of their abilities. Of the possible subsets of problems whose total point values add up to 100, a knapsack algorithm would determine which subset gives each student the highest possible score. The most common problem being solved is the 0-1 knapsack problem, which restricts the number x i {displaystyle x_{i}} of copies of each kind of item to zero or one. Given a set of n {displaystyle n} items numbered from 1 up to n {displaystyle n} , each with a weight w i {displaystyle w_{i}} and a value v i {displaystyle v_{i}} , along with a maximum weight capacity W {displaystyle W} , Here x i {displaystyle x_{i}} represents the number of instances of item i {displaystyle i} to include in the knapsack. 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. The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number x i {displaystyle x_{i}} of copies of each kind of item to a maximum non-negative integer value c {displaystyle c} :