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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.
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.
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 can be solved by dynamic programming in pseudo-polynomial time: (), where m is the number of inputs and V is the number of different possible values. To get a polynomial-time algorithm, we can solve the knapsack problem approximately, using input rounding.
When the number of bins is restricted to 1 and each item is characterized by both a volume and a value, the problem of maximizing the value of items that can fit in the bin is known as the knapsack problem. A variant of bin packing that occurs in practice is when items can share space when packed into a bin.
The separation oracle for the dual LP can be implemented by solving the knapsack problem with sizes s and values y: if the optimal solution of the knapsack problem has a total value at most 1, then y is feasible; if it is larger than 1, than y is not feasible, and the optimal solution of the knapsack problem identifies a configuration for which ...
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.
For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program. The knapsack problem has well-known methods to solve it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be ...