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The most common problem being solved is the 0-1 knapsack problem, which restricts the number of copies of each kind of item to zero or one. Given a set of items numbered from 1 up to , each with a weight and a value , along with a maximum weight capacity ,
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. For this reason, many special cases and generalizations have been examined.
It is a special case of the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem , a general case of partition in which, given the available denominations of an infinite set of coins, the objective is to find out the number of possible ways of making a change ...
The quadratic knapsack problem (QKP), first introduced in 19th century, [1] is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as ...
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.
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The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
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