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| In [[theoretical computer science]], the '''continuous knapsack problem''' (also known as the '''fractional knapsack problem''') is an [[algorithm]]ic problem in [[combinatorial optimization]] in which the goal is to fill a container (the "knapsack") with fractional amounts of different materials chosen to maximize the value of the selected materials.<ref name="gt">{{citation|title=Algorithm Design: Foundations, Analysis, and Internet Examples|first1=Michael T.|last1=Goodrich|author1-link=Michael T. Goodrich|first2=Roberto|last2=Tamassia|author2-link=Roberto Tamassia|publisher=John Wiley & Sons|year=2002|contribution=5.1.1 The Fractional Knapsack Problem|pages=259–260}}.</ref><ref name="co">{{citation|title=Combinatorial Optimization: Theory and Algorithms|volume=21|series=Algorithms and Combinatorics|first1=Bernhard|last1=Korte|author1-link=Bernhard Korte|first2=Jens|last2=Vygen|publisher=Springer|year=2012|isbn=9783642244889|contribution=17.1 Fractional Knapsack and Weighted Median|pages=459–461|url=http://books.google.com/books?id=8535vmYbLGYC&pg=PA459}}.</ref> It resembles the classic [[knapsack problem]], in which the items to be placed in the container are indivisible; however, the continuous knapsack problem may be solved in [[polynomial time]] whereas the classic knapsack problem is [[NP-hard]].<ref name="gt"/>
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| ==Problem definition==
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| An instance of the either the continuous or classic knapsack problems may be specified by the numerical capacity ''W'' of the knapsack, together with a collection of materials, each of which has two numbers associated with it: the weight ''w<sub>i</sub>'' of material that is available to be selected and the value per unit weight ''v<sub>i</sub>'' of that material. The goal is to choose an amount ''x<sub>i</sub>'' ≤ ''w<sub>i</sub>'' of each material, subject to the capacity constraint
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| :<math>\sum_i x_i\le W</math>
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| and maximizing the total benefit
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| :<math>\sum_i x_i v_i</math>.
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| In the classic knapsack problem, each of the amounts ''x<sub>i</sub>'' must be either zero or ''w<sub>i</sub>''; the continuous knapsack problem differs by allowing ''x<sub>i</sub>'' to range continuously from zero to ''w<sub>i</sub>''.<ref name="gt"/>
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| Some formulations of this problem rescale the variables ''x<sub>i</sub>'' to be in the range from 0 to 1
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| ==Solution technique==
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| The continuous knapsack problem may be solved by a [[greedy algorithm]], first published in 1957 by [[George Dantzig]],<ref name="co"/><ref>{{citation
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| | last = Dantzig | first = George B. | authorlink = George Dantzig
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| | doi = 10.1287/opre.5.2.266
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| | journal = Operations Research
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| | mr = 0089098
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| | pages = 266–277
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| | title = Discrete-variable extremum problems
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| | volume = 5
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| | year = 1957}}.</ref> that considers the materials in sorted order by their values per unit weight. For each material, the amount ''x<sub>i</sub>'' is chosen to be as large as possible:
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| *If the sum of the choices made so far equals the capacity ''W'', then the algorithm sets ''x<sub>i</sub>'' = 0.
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| *If the difference ''d'' between the sum of the choices made so far and ''W'' is smaller than ''w<sub>i</sub>'', then the algorithm sets ''x<sub>i</sub>'' = ''d''.
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| *In the remaining case, the algorithm chooses ''x<sub>i</sub>'' = ''w<sub>i</sub>''.
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| Because of the need to sort the materials, this algorithm takes time ''O''(''n'' log ''n'') on inputs with ''n'' materials.<ref name="gt"/><ref name="co"/> However, by adapting an algorithm for finding [[weighted median]]s, it is possible to solve the problem in time ''O''(''n'').<ref name="co"/>
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| ==References==
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| {{reflist}}
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| [[Category:Combinatorial optimization]]
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