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The knapsack problem is an optimization problem used to illustrate both problem and solution. It derives its name from a scenario where one is constrained in the number of items that can be placed inside a fixed-size knapsack. Given a set of items with specific weights and values, the aim is to get as much value into the knapsack as possible given the weight constraint of the knapsack.
The knapsack problem is an example of a combinational optimization problem, a topic in mathematics and computer science about finding the optimal object among a set of objects. This is a problem that has been studied for more than a century and is a commonly used example problem in combinatorial optimization, where there is a need for an optimal object or finite solution where an exhaustive search is not possible. The problem can be found real-world scenarios like resource allocation in financial constraints or even in selecting investments and portfolios. It also can be found in fields such as applied mathematics, complexity theory, cryptography, combinatorics and computer science. It is easily the most important problem in logistics.
In the knapsack problem, the given items have two attributes at minimum – an item’s value, which affects its importance, and an item’s weight or volume, which is its limitation aspect. Since an exhaustive search is not possible, one can break the problems into smaller sub-problems and run it recursively. This is called an optimal sub-structure. This deals with only one item at a time and the current weight still available in the knapsack. The problem solver only needs to decide whether to take the item or not based on the weight that can still be accepted. However, if it is a program, re-computation is not independent and would cause problems. This is where dynamic programming techniques can be applied. Solutions to each sub-problem are stored so that the computation would only need to happen once.