Chapter 4: Problem 39

Prove that the greedy approach to the Fractional Knapsack problem yields an optimal solution.

Short Answer

Expert verified

The proof shows that replacing any non-greedy item with a greedy item in a supposedly optimal solution increases the total value in the knapsack. This leads to the conclusion that the greedy approach is indeed optimal for the Fractional Knapsack problem.

Step by step solution

Understand the Problem

The Fractional Knapsack problem is basically about maximizing the total value of the items picked without exceeding the weight capacity of the knapsack. Each item has a certain weight and value. The idea of the greedy approach in this context is to take the item with the highest value/weight ratio first, then the next highest and so on until the knapsack is full.

Setup of the proof

To prove that this approach is optimal, we assume that there's an optimal solution different from the Greedy solution, and then try to derive a contradiction from this assumption.

The Contradiction

Let's assume an item in the Greedy solution that is not in our supposed optimal solution (a non-greedy item). This non-greedy item will have a smaller value/weight ratio compared to the items in the Greedy solution. If we replace this non-greedy item (or part of it) in our supposed optimal solution with an item from the Greedy solution, we'll find that we've increased the total value in the knapsack, that contradicts our assumption that the non-greedy solution was optimal.

Conclusion of the proof

Since replacing the non-greedy item with a greedy item leads to a better solution, it proves that the greedy approach is optimal. If there was a better (non-greedy) solution, we could always make it at least as good by replacing the non-greedy parts with the greedy parts.

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Prove that the greedy approach to the Fractional Knapsack problem yields an optimal solution.

Short Answer

Step by step solution

Understand the Problem

Setup of the proof

The Contradiction

Conclusion of the proof

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