Continuous knapsack problem


In theoretical computer science, the continuous knapsack problem is an algorithmic problem in combinatorial optimization in which the goal is to fill a container with fractional amounts of different materials chosen to maximize the value of the selected materials. 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. 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 wi of material that is available to be selected and the total value vi of that material. The goal is to choose an amount xi ≤ wi of each material, subject to the capacity constraint
and maximizing the total benefit
In the classic knapsack problem, each of the amounts xi must be either zero or wi; the continuous knapsack problem differs by allowing xi to range continuously from zero to wi.
Some formulations of this problem rescale the variables xi to be in the range from 0 to 1. In this case the capacity constraint becomes
and the goal is to maximize the total benefit

Solution technique

The continuous knapsack problem may be solved by a greedy algorithm, first published in 1957 by George Dantzig, that considers the materials in sorted order by their values per unit weight. For each material, the amount xi is chosen to be as large as possible:
Because of the need to sort the materials, this algorithm takes time O on inputs with n materials. However, by adapting an algorithm for finding weighted medians, it is possible to solve the problem in time O.