Cs 598csc: Approximation Algorithms 1 the Knapsack Problem 1.1 Problem Description 1.2 a Greedy Algorithm

In the Knapsack problem we are given a knapsack capacity B, and set N of n items. Each item i has a given size si ≥ 0 and a profit pi ≥ 0. Given a subset of the items A ⊆ N , we define two functions, s(A) = ∑ i∈A si and p(A) = ∑ i∈A pi, representing the total size and profit of the group of items respectively. The goal is to choose a subset of the items, A, such that s(A) ≤ B and p(A) is maximized. We will assume every item has size at most B. It is not difficult to see that if all the profits are 1, the natural greedy algorithm of sorting the items by size and then taking the smallest items will give an optimal solution. Assuming the profits and sizes are integral, we can still find an optimal solution to the problem relatively quickly using dynamic programming in either O(nB) or O(nP ) time, where P = ∑n i=1 pi. Finding the details of these algorithms was given as an exercise in the practice homework. While these algorithms appear to run in polynomial time, it should be noted that B and P can be exponential in the size of the input assuming the numbers in the input are not written in unary. We call these pseudo-polynomial time algorithms as their running times are polynomial only when numbers in the input are given in unary.