how to proof fractional knapsack algorithm

Daniel Parker

2 min read

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31-01-2025

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The fractional knapsack problem is a classic optimization problem where you have a knapsack with a weight capacity and a set of items, each with a weight and a value. The goal is to maximize the total value of items you put in the knapsack without exceeding its weight capacity. Unlike the 0/1 knapsack problem, you can take fractions of items. This allows for a simpler, greedy approach to finding the optimal solution. This article will explain how to prove the correctness of the greedy algorithm used to solve this problem.

Understanding the Greedy Approach

The greedy approach to the fractional knapsack problem involves sorting the items by their value-to-weight ratio (V/W) in descending order. Then, you iteratively add items to the knapsack, taking as much of each item as possible without exceeding the knapsack's capacity. This continues until the knapsack is full or all items are considered.

Proof of Correctness

The correctness of this greedy approach can be proven using an exchange argument. The core idea is to show that if a solution isn't sorted by the V/W ratio, we can always improve it by swapping items until it is.

1. Optimal Solution Exists:

First, let's establish that an optimal solution exists. Since we have a finite number of items and a finite knapsack capacity, there's a finite number of possible combinations. Therefore, an optimal combination maximizing the total value must exist.

2. Exchange Argument:

Assume we have an optimal solution S where at least two items, i and j, are included in a way that violates the V/W ratio order (item i has a lower V/W ratio than item j but is included more than item j).

3. The Swap:

Now, let's consider swapping a small amount of item i with an equal weight of item j. Let's say Δw is the weight being swapped. The change in value will be:

ΔV = (Vj/Wj - Vi/Wi) * Δw

Since Vj/Wj > Vi/Wi, ΔV will be positive. This means the new solution with the swap will have a higher total value than the initial solution S. This contradicts our assumption that S was optimal.

4. Iterative Improvement:

We can repeat this swap process for any pair of items violating the V/W ratio order. With each swap, we increase the total value. Because the number of items is finite, this process will eventually converge to a solution where no such swaps can improve the total value. This solution will be sorted by the V/W ratio, demonstrating that the greedy approach indeed leads to an optimal solution.

5. Formal Proof (Mathematical Induction):

A more formal proof can be constructed using mathematical induction. This would involve:

• Base Case: Showing that the greedy approach works for a knapsack with only one item.
• Inductive Step: Assuming the greedy approach works for a knapsack with k items, proving it works for a knapsack with k+1 items. This step would involve analyzing the inclusion of the (k+1)th item and leveraging the exchange argument outlined above.

Conclusion

The greedy approach to the fractional knapsack problem is correct and guarantees finding the optimal solution. This is due to the fact that if a solution isn't ordered by the value-to-weight ratio, we can always find a better solution by exchanging portions of items. This exchange argument, along with a formal proof using mathematical induction, solidifies the algorithm's correctness. The simplicity and efficiency of the greedy algorithm make it a valuable tool in various optimization problems.