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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 n {\displaystyle n} items numbered from 1 up to n {\displaystyle n} , each with a weight w i {\displaystyle w_{i}} and a value v i {\displaystyle v_{i}} , along with a maximum weight capacity ...
Both the bounded and unbounded variants admit an FPTAS (essentially the same as the one used in the 0-1 knapsack problem). If the items are subdivided into k classes denoted N i {\displaystyle N_{i}} , and exactly one item must be taken from each class, we get the multiple-choice knapsack problem :
The knapsack problem, [17] [18] as well as some of its variants: 0-1 knapsack problem. [19] Unbounded knapsack problem. [20] Multi-dimensional knapsack problem with Delta-modular constraints. [21] Multi-objective 0-1 knapsack problem. [22] Parametric knapsack problem. [23] Symmetric quadratic knapsack problem. [24]
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 ...
For example, bin packing is strongly NP-complete while the 0-1 Knapsack problem is only weakly NP-complete. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming.
The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling
3SUM – Problem in computational complexity theory Merkle–Hellman knapsack cryptosystem – one of the earliest public key cryptosystems invented by Ralph Merkle and Martin Hellman in 1978. The ideas behind it are simpler than those involving RSA, and it has been broken Pages displaying wikidata descriptions as a fallback
In the classic knapsack problem, each of the amounts x i must be either zero or w i; the continuous knapsack problem differs by allowing x i to range continuously from zero to w i. [1] Some formulations of this problem rescale the variables x i to be in the range from 0 to 1.