0-1 Knapsack Problem

• The thief wants to steal n items
• The ith item weighs w_i and has value v_i
• Take the most valuable load, limit of W pounds
• This is called the 0-1 version of the problem because there are no fractions.
  The thief must take the whole item or leave it behind.
• Both the 0-1 Knapsack Problem and the Fractional Knapsack Problem have optimal substructure. If we remove item j from the optimal load, the remaining problem must be optimal for the remaining W - w_j pounds for the overall solution to be optimal.
• The greedy solution for the Fractional Knapsack Problem does not work here.
  

Example:

Item	Value	Weight	Val/Weight
1	60	10	6
2	100	20	5
3	120	30	4

W = 50

The greedy solution would select 1, leaving 40 pounds.

At this point if we pick items 1 and 2 the total value is 160.

If we pick items 1 and 3 the total value is 180.

The optimal solution is to pick items 2 and 3. The total value of this solution is 220.