One theme in research literature is to identify what the 'hard' instances of the knapsack problem look like, or viewed another way, to identify what properties of instances in practice might make them more amenable than their worst-case NP-complete behaviour suggests. Merkle-Hellman and several similar algorithms were later broken, because the particular subset sum problems they produced were in fact solvable by polynomial-time algorithms. These attempts typically used some group other than the integers. There have been attempts to use subset sum as the basis for public key cryptography systems, such as the Merkle-Hellman knapsack cryptosystem. The subset sum version of the knapsack problem is commonly known as one of Karp's 21 NP-complete problems.

many cases that arise in practice, and 'random instances' from some distributions, can nonetheless be solved exactly.the problem is NP-complete to solve exactly, thus it is expected that no algorithm can be both correct and fast (polynomial-time) on all cases.there is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine.there is a pseudo-polynomial time algorithm using dynamic programming.The knapsack problem is interesting from the perspective of computer science because If multiple knapsacks are allowed, the problem is better thought of as the bin packing problem. In the field of cryptography the term knapsack problem is often used to refer specifically to the subset sum problem. Notice that in this special case, the problem is equivalent to this: given a set of nonnegative integers, does any subset of it add up to exactly W? Or, if negative weights are allowed and W is chosen to be zero, the problem is: given a set of integers, does any subset add up to exactly 0? This special case is called the subset sum problem. for each kind of item, the weight equals the value.Of particular interest is the special case of the problem with these properties: The unbounded knapsack problem places no upper bound on the number of copies of each kind of item. Mathematically the bounded knapsack problem can be formulated as: The bounded knapsack problem restricts the number of copies ofД®ach kind of item to a maximum integer value. Mathematically the 0-1-knapsack problem can be formulated as:

The most common formulation of the problem is the 0-1 knapsack problem, which restricts the number x i of copies of each kind of item The maximum weight that we can carry in the bag is W. To simplify the representation, we can also assume that the items are listed in increasing order of weight. We usually assume that all values and weights are nonnegative. In the following, we have n kinds of items, 1 through n.Д®ach kind of item i has a value v i and a weight w i.

5 Dominance relations to simplify the resolution of the unbounded knapsack problem.

The decision problem form of the knapsack problem is the question 'can a value of at least V be achieved without exceeding the weight W?' A similar problem also appears in combinatorics, complexity theory, cryptography and applied mathematics. The problem often arises in resource allocation with financial constraints. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most useful items.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than a given limit and the total value is as large as possible. (Solution: if any number of each box is available, then three yellow boxes and three grey boxes if only the shown boxes are available, then all but the green box.) Modeling the shapes and sizes would instead constitute a packing problem. Example of a one-dimensional (constraint) knapsack problem: which boxes should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15В kg? A multiple constrained problem could consider both the weight and volume of the boxes.

Knapsack definition