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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 and applied mathematics. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. It is not known how the name "knapsack problem" originated, though the problem was referred to as such in the early works of mathematician Tobias Dantzig (1884–1956),〔Dantzig, Tobias. Numbers: The Language of Science, 1930.〕 suggesting that the name could have existed in folklore before a mathematical problem had been fully defined.〔Kellerer, Pferschy, and Pisinger 2004, p. 3〕 ==Applications== A 1998 study of the (Stony Brook University Algorithm Repository ) showed that, out of 75 algorithmic problems, the knapsack problem was the 18th most popular and the 4th most needed after kd-trees, 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,〔Kellerer, Pferschy, and Pisinger 2004, p. 449〕 seating contest of investments and portfolios,〔Kellerer, Pferschy, and Pisinger 2004, p. 461〕 seating contest of assets for asset-backed securitization,〔Kellerer, Pferschy, and Pisinger 2004, p. 465〕 and generating keys for the Merkle–Hellman〔Kellerer, Pferschy, and Pisinger 2004, p. 472〕 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—i.e. different questions are worth different 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Knapsack problem」の詳細全文を読む スポンサード リンク
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