GATE 2018 CS – Question 58
Consider the weights and values of items listed below. Note that there is only one unit of each item.
| Item number | Weight (in Kgs) | Value (in Rupees) |
|---|---|---|
| 1 | 10 | 60 |
| 2 | 7 | 28 |
| 3 | 4 | 20 |
| 4 | 2 | 24 |
The task is to pick a subset of these items such that their total weight is no more than 11 Kgs and their total value is maximized. Moreover, no item may be split. The total value of items picked by an optimal algorithm is denoted by $V_{opt}$. A greedy algorithm sorts the items by their value-to-weight ratios in descending order and packs them greedily, starting from the first item in the ordered list. The total value of items picked by the greedy algorithm is denoted by $V_{greedy}$.
The value of $V_{opt}-V_{greedy}$ is ________.
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Show answer and explanation
Correct answer: 16
Explanation
The ratios are $6,4,5,12$, so greedy takes item 4 (weight 2, value 24) and item 3 (weight 4, value 20) and then cannot fit item 1 or 2: $V_{greedy}=44$. The best choice is item 1 alone (value 60), so $V_{opt}=60$ and the difference is 16.