Knapsack Problem or Rucksack Problem

KNAPSACK PROBLEM

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. 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 problem often arises in resource allocation with financial constraints. A similar problem also appears in combinatorics, complexity theory, cryptography and applied mathematics.

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?"

E.g.

A thief enters a store and sees the following items:

His Knapsack holds 4 pounds.

What should he steal to maximize profit?

Fractional Knapsack Problem

Thief can take a fraction of an item.

GREEDY TECHNIQUES

The greedy approach suggests constructing a solution through a sequence of steps, each expanding a partially constructed solution obtained so far, until a complete solution to the problem is reached. On each step—and this is the central point of this technique—the choice made must be

•        Feasible, i.e., it has to satisfy the problem‘s constraints.

•        Locally optimal, i.e., it has to be the best local choice among all feasible choices available on that step.

•        Irrevocable, i.e., once made, it cannot be changed on subsequent steps of the algorithm.