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A 1999 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems related to the field of combinatorial algorithms and algorithm engineering, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem.
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 algorithm can be made much more effective by first sorting the list of items into decreasing order (sometimes known as the first-fit decreasing algorithm), although this still does not guarantee an optimal solution and for longer lists may increase the running time of the algorithm. It is known, however, that there always exists at least ...
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 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 ...
A knapsack auction is an auction in which several identical items are sold, and there are several bidders with different valuations interested in different amounts of items. The goal is to choose a subset of the bidders with a total demand, at most, the number of items and, subject to that, a maximum total value.
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.
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 ...