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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 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 ...
The bin packing problem can also be seen as a special case of the cutting stock problem. 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.
Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling; Numerical 3-dimensional matching [3]: SP16 Open-shop scheduling; Partition problem [2] [3]: SP12 Quadratic assignment problem [3]: ND43 Quadratic programming (NP-hard in some cases, P if convex)
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 ...
Indeed, this problem does not have an FPTAS unless P=NP. The same is true for the two-dimensional knapsack problem. The same is true for the multiple subset sum problem: the quasi-dominance relation should be: s quasi-dominates t iff max(s 1, s 2) ≤ max(t 1, t 2), but it is not preserved by transitions, by the same example as above. 2.
Karp's 21 problems are shown below, many with their original names. The nesting indicates the direction of the reductions used. For example, Knapsack was shown to be NP-complete by reducing Exact cover to Knapsack. Satisfiability: the boolean satisfiability problem for formulas in conjunctive normal form (often referred to as SAT)
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.