List of knapsack problems
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. Common to all versions are a set of "n" items, with each item formula_1 having an associated profit "p", weight "w". The binary decision variable "x" is used to select the item. The objective is to pick some of the items, with maximal total profit, while obeying that the maximum total weight of the chosen items must not exceed "W". Generally, these coefficients are scaled to become  integers, and they are almost always assumed to be positive. The knapsack problem in its most basic form: Direct generalizations. One common variant is that each item can be chosen multiple times. The bounded knapsack problem specifies, for each item "j", an upper bound "u" (which may be a positive integer, or infinity) on the number of times item "j" can be selected: The unbounded knapsack problem (sometimes called the integer knapsack problem) does not put any upper bounds on the number of times an item may be selected: The unbounded variant was shown to be NP-complete in 1975  by Lueker. 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 formula_2, and exactly one item must be taken from each class, we get the multiple-choice knapsack problem: If for each item the profit and weight are equal, we get the subset sum problem (often the corresponding decision problem is given instead): If we have "n" items and "m" knapsacks with capacities formula_3, we get the multiple knapsack problem: As a special case of the multiple knapsack  problem, when the profits are equal to weights and all bins have the same capacity, we can have multiple subset sum problem. Quadratic knapsack problem: Set-Union Knapsack Problem: SUKP is defined by Kellerer et al (on page 423) as follows: Given a set of formula_4 items formula_5 and a set of formula_6 so-called elements formula_7, each item formula_8 corresponds to a subset formula_9 of the element set formula_10. The items formula_8 have non-negative profits formula_12, formula_13, and the elements formula_14 have non-negative weights formula_15, formula_16. The total weight of a set of items is given by  the total weight of the elements of the union of the corresponding element sets. The objective is to find a subset of the items with total weight not exceeding the knapsack capacity and maximal profit. Multiple constraints. If there is more than one constraint (for example, both a volume limit and a weight limit, where the volume and weight of each item are not related), we get the multiple-constrained knapsack problem, multidimensional knapsack problem, or "m"-dimensional knapsack problem. (Note, "dimension" here does not refer to the shape of any items.) This has 0-1, bounded, and unbounded variants; the  unbounded one is shown below. The 0-1 variant (for any fixed formula_17) was shown to be NP-complete around 1980 and more strongly, has no FPTAS unless P=NP. The bounded and unbounded variants (for any fixed formula_17) also exhibit the same hardness. For any fixed formula_17, these problems do admit a pseudo-polynomial time algorithm (similar to the one for basic knapsack) and a PTAS. Knapsack-like problems. If all the profits are 1, we will try to maximize the number of items which would not exceed the knapsack capacity: If we have a number of containers  (of the same size), and we wish to pack all "n" items in as few containers as possible, we get the bin packing problem, which is modelled by having indicator variables formula_20 container "i" is being used: The cutting stock problem is identical to the bin packing problem, but since practical instances usually have far fewer types of items, another formulation is often used. Item "j" is needed "B" times, each "pattern" of items which fit into a single knapsack have a variable, "x" (there are "m" patterns), and pattern "i" uses item "j" "b" times: If, to the multiple  choice knapsack problem, we add the constraint that each subset is of size "n" and remove the restriction on total weight, we get the assignment problem, which is also the problem of finding a maximal bipartite matching: In the Maximum Density Knapsack variant there is an initial weight formula_21, and we maximize the density of selected of items which do not violate the capacity constraint: Although less common than those above, several other knapsack-like problems exist, including: The last three of these are discussed in Kellerer et al's reference work, "Knapsack Problems".