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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.
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.
Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, [3] selection of investments and portfolios, [4] selection of assets for asset-backed securitization, [5] and generating keys for the Merkle–Hellman [6] and other knapsack cryptosystems.
The problem of fractional knapsack with penalties was introduced by Malaguti, Monaci, Paronuzzi and Pferschy. [44] They developed an FPTAS and a dynamic program for the problem, and they showed an extensive computational study comparing the performance of their models.
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.
de Bruijn's theorem: A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p, q, r (i.e., the box is a multiple of the brick.) [15]
Note: consider In the 2-weighted knapsack problem, where each item has two weights and a value, and the goal is to maximize the value such that the sum of squares of the total weights is at most the knapsack capacity: (,) + (,). We could solve it using a similar DP, where each state is (current weight 1, current weight 2, value).
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 ...