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: w j = p j.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. The bounded knapsack problem restricts the number x j of copies ofeach kind of item to a maximum integer value b j.Mathematically the bounded knapsack problem can be formulated as: maximize subject to The most common formulation of the problem is the 0-1 knapsack problem, which restricts the number x j of copies of each kind of itemto zero or one.Mathematically the 0-1-knapsack problem can be formulated as: maximize subject to We usually assume that all values and weights are nonnegative.The maximum weight that we can carry in the bag is W. Each kind of item j has a value p j and a weight w j. In the following, we have n kinds of items, 1 through n. 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. Sack-race (n.) is attested from 1805.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 15 kg? A multiple constrained problem could consider both the weight and volume of the boxes. The slang meaning 'bunk, bed' is by 1825, originally nautical, hence many slang phrases, originally nautical, such as sack duty 'sleep ' the verb meaning 'go to bed' is recorded from 1946. The baseball slang sense of 'a base' is attested from 1913. In English, the meaning 'a sack or sack material used as an article of clothing' as a token of penitence or mourning is from c. The wide spread of this word for 'a bag' probably is due to the incident in the Biblical story of Joseph in which a sack of corn figures (Genesis xliv). 'large oblong bag,' Middle English sak, from Old English sacc (West Saxon), sec (Mercian), sГ¦c (Old Kentish) 'large cloth bag,' also 'sackcloth,' from Proto-Germanic *sakkiz (source also of Middle Dutch sak, Old High German sac, Old Norse sekkr, but Gothic sakkus probably is directly from Greek), an early borrowing from Latin saccus (also source of Old French sac, Spanish saco, Italian sacco), from Greek sakkos 'bag (made of goat hair) sieve burlap, large burlap cloak,' which is from Semitic (compare Hebrew, Phoenician saq 'sack, cloth of hair, bag, mourning-dress').

Knapsack definition