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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.
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.
One variation of this problem assumes that the people making change will use the "greedy algorithm" for making change, even when that requires more than the minimum number of coins. Most current currencies use a 1-2-5 series , but some other set of denominations would require fewer denominations of coins or a smaller average number of coins to ...
The budgeting method most common in practice is a greedy solution to a variant of the knapsack problem: the projects are ordered by decreasing order of the number of votes they received, and selected one-by-one until the budget is exhausted. Alternatively, if the number of projects is sufficiently small, the knapsack problem may be solved ...
The continuous knapsack problem may be solved by a greedy algorithm, first published in 1957 by George Dantzig, [2] [3] that considers the materials in sorted order by their values per unit weight. For each material, the amount x i is chosen to be as large as possible:
For the problem variant in which not every item must be assigned to a bin, there is a family of algorithms for solving the GAP by using a combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for the GAP. [3] Using any -approximation algorithm ALG for the knapsack problem, it is possible to ...
Upload file; Search. Search. Appearance. ... Continuous knapsack problem; Criss-cross algorithm; Cut (graph theory) ... Greedy randomized adaptive search procedure; H.
However, if is superincreasing, meaning that each element of the set is greater than the sum of all the numbers in the set lesser than it, the problem is "easy" and solvable in polynomial time with a simple greedy algorithm. In Merkle–Hellman, decrypting a message requires solving an apparently "hard" knapsack problem.