Continuous knapsack problem

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In theoretical computer science, the continuous knapsack problem (also known as the fractional knapsack problem) is an algorithmic 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.^[1]^[2] 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.^[1]

Problem definition

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_i of material that is available to be selected and the value per unit weight v_i of that material. The goal is to choose an amount x_i ≤ w_i of each material, subject to the capacity constraint

$\sum _{i}x_{i}\leq W$

and maximizing the total benefit

$\sum _{i}x_{i}v_{i}$ .

In the classic knapsack problem, each of the amounts x_i must be either zero or w_i; the continuous knapsack problem differs by allowing x_i to range continuously from zero to w_i.^[1] Some formulations of this problem rescale the variables x_i to be in the range from 0 to 1

Solution technique

The continuous knapsack problem may be solved by a greedy algorithm, first published in 1957 by George Dantzig,^[2]^[3] that considers the materials in sorted order by their values per unit weight. For each material, the amount x_i is chosen to be as large as possible:

If the sum of the choices made so far equals the capacity W, then the algorithm sets x_i = 0.
If the difference d between the sum of the choices made so far and W is smaller than w_i, then the algorithm sets x_i = d.
In the remaining case, the algorithm chooses x_i = w_i.

Because of the need to sort the materials, this algorithm takes time O(n log n) on inputs with n materials.^[1]^[2] However, by adapting an algorithm for finding weighted medians, it is possible to solve the problem in time O(n).^[2]

References

↑ 1.0 1.1 1.2 1.3 Goodrich, Michael T.; Tamassia, Roberto (2002), "5.1.1 The Fractional Knapsack Problem", Algorithm Design: Foundations, Analysis, and Internet Examples, John Wiley & Sons, pp. 259–260 .
↑ 2.0 2.1 2.2 2.3 Korte, Bernhard; Vygen, Jens (2012), "17.1 Fractional Knapsack and Weighted Median", Combinatorial Optimization: Theory and Algorithms, Algorithms and Combinatorics 21, Springer, pp. 459–461, ISBN 9783642244889 .
↑ Dantzig, George B. (1957), "Discrete-variable extremum problems", Operations Research 5: 266–277, doi:10.1287/opre.5.2.266, MR 0089098 .

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