## The Knapsack Problem(s)

In this section, we will discuss the standard **knapsack problem**, also known as the 0-1 knapsack problem, which is known to be NP-complete, and thereby does not allow us to have any polynomial-time solution. Then, we will turn our discussion toward a version of the knapsack problem called the **fractional knapsack problem**, which can be solved using a greedy approach. Our focus in this section is to demonstrate how even subtle differences between how a problem is defined can lead to large changes in the solution strategies.

### The Knapsack Problem

Suppose you are given a set of objects, *O = {O**1**, O**2**, …, O**n**}*, with each having a certain weight, *W**i*, and a value of *V**i**.* You are also given a bag (or a knapsack) that can carry only a total weight of T units. Now, say you are tasked with finding out about a set of objects to keep in your bag so that the total weight is less than or equal to T, and the total value of the objects is the maximum it can possibly be.

A real-world example...