Continuous knapsack problem

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] It is a classic example of how a seemingly small change in the formulation of a problem can have a large impact on its computational complexity.

Problem definition

An instance of 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 total value v_i of that material. The goal is to choose an amount x_i ≤ w_i of each material, subject to the capacity constraint $${\displaystyle \sum _{i}x_{i}\leq W}$$ and maximizing the total benefit $${\displaystyle \sum _{i}x_{i}v_{i}.}$$ 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. In this case the capacity constraint becomes $${\displaystyle \sum _{i}x_{i}w_{i}\leq W,}$$ and the goal is to maximize the total benefit $${\displaystyle \sum _{i}x_{i}v_{i}.}$$

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. 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. Korte, Bernhard; Vygen, Jens (2012), "17.1 Fractional Knapsack and Weighted Median", Combinatorial Optimization: Theory and Algorithms, Algorithms and Combinatorics, vol. 21, Springer, pp. 459–461, ISBN 9783642244889.
3. Dantzig, George B. (1957), "Discrete-variable extremum problems", Operations Research, 5: 266–277, doi:10.1287/opre.5.2.266, MR 0089098.