GATE CS 2018 Set 1 — Question 58
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Algorithms → Dynamic Programming → 0/1 Knapsack
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Question
Consider the weights and values of items listed below. Note that there is only one unit of each item.
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 . 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 .The value of is ____________.
| Item number | Weight (in Kgs) | Value (in Rupees) |
|---|---|---|
| 1 | 10 | 60 |
| 2 | 7 | 28 |
| 3 | 4 | 20 |
| 4 | 2 | 24 |
Correct answer
16 to 16
Solution
1.Optimal Solution (): This is a 0/1 Knapsack problem with capacity . We evaluate subsets of items:
- {Item 1}: Weight = 10, Value = 60
- {Item 2, Item 3}: Weight = 7 + 4 = 11, Value = 28 + 20 = 48
- {Item 2, Item 4}: Weight = 7 + 2 = 9, Value = 28 + 24 = 52
- {Item 3, Item 4}: Weight = 4 + 2 = 6, Value = 20 + 24 = 44
2.Greedy Solution (): Sort items by value-to-weight ratio ():
- Item 1:
- Item 2:
- Item 3:
- Item 4:
Greedy selection process:
- Pick Item 4: Weight = 2, Value = 24. Remaining capacity = 9.
- Try Item 1: Weight = 10. Exceeds remaining capacity (9). Skip.
- Try Item 3: Weight = 4, Value = 20. Remaining capacity = 5.
- Try Item 2: Weight = 7. Exceeds remaining capacity (5). Skip.
3.Calculation: .
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