The Fractional Knapsack Problem is a classic optimization problem in computer science and mathematics. The problem involves selecting items of certain values and weights to maximize the total value while ensuring that the total weight does not exceed a given capacity.
Here's a step-by-step explanation of how the Fractional Knapsack Problem can be solved -
Given a set of items, each with a value (vi) and weight (wi), and a knapsack with a maximum capacity (W), calculate the value-to-weight ratio for each item by dividing the value of each item by its weight.
Sort the items in descending order based on their value-to-weight ratios. This step allows us to consider the most valuable items first.
Initialize the maximum value of the knapsack (maxValue) and an empty fractional solution (selectedItems).
Iterate through the sorted items. For each item, check if it can be fully included in the knapsack without exceeding the capacity (W). If it can, add the item's value to maxValue and update the capacity by subtracting the item's weight from W. Also, add the entire item to the selectedItems list.
If an item cannot be fully included due to capacity limitations, calculate the fraction of the item that can fit into the remaining capacity. Add the fraction of the item's value to maxValue and update the selectedItems list with the fractional item.
Repeat steps 4 and 5 until the knapsack reaches its maximum capacity (W) or all items have been considered.
Return the maximum value (maxValue) and the selectedItems list, which represents the optimal solution.
The Fractional Knapsack Problem differs from the 0/1 Knapsack Problem, where items cannot be divided but must be either taken entirely or left out. In the Fractional Knapsack Problem, items can be divided and included in fractions, hence the name "fractional." This flexibility allows for obtaining non-integer solutions.
The Fractional Knapsack Problem can be efficiently solved using a greedy algorithm, as outlined in the steps above. The greedy approach selects the items with the highest value-to-weight ratios first, making the locally optimal choice at each step.
Greedy Strategy: The Fractional Knapsack Problem can be solved optimally using a greedy strategy due to its unique property. Unlike the 0/1 Knapsack Problem, the Fractional Knapsack Problem allows for taking fractions of items, which enables the greedy approach of selecting items with the highest value-to-weight ratios at each step. This strategy guarantees an optimal solution because it maximizes the value per unit of weight.
Complexity - The greedy algorithm for the Fractional Knapsack Problem has a time complexity of O(n log n), where n is the number of items. This complexity arises from the initial sorting step based on value-to-weight ratios. After sorting, the algorithm iterates through the sorted list once to determine the optimal solution.
Continuous vs. Discrete - The Fractional Knapsack Problem deals with continuous values, meaning items can be divided into fractions. In contrast, the 0/1 Knapsack Problem deals with discrete values, where items are either included entirely or excluded. This distinction makes the Fractional Knapsack Problem more flexible and allows for real-life scenarios where dividing items is possible.
Real-life Applications - The Fractional Knapsack Problem has numerous practical applications. It can be used in resource allocation problems, such as selecting stocks in an investment portfolio, maximizing the utilization of a limited resource, or optimizing the loading of cargo onto a vehicle with weight and volume constraints.
Dynamic Programming - While the greedy algorithm is efficient and guarantees an optimal solution for the Fractional Knapsack Problem, it is not suitable for the 0/1 Knapsack Problem. The 0/1 Knapsack Problem requires a dynamic programming approach to find the optimal solution due to its restriction on taking items in fractions. Dynamic programming breaks down the problem into smaller subproblems and utilizes memoization to avoid redundant calculations.
Pseudopolynomial Time - The Fractional Knapsack Problem has a pseudopolynomial time complexity, meaning the running time is polynomial in the numeric value of the input rather than the input size. The time complexity depends on the maximum value of the items rather than the number of items. This distinction is important when considering the efficiency of algorithms for solving the problem.
Remember, the Fractional Knapsack Problem is a versatile and widely studied optimization problem that offers practical applications and can be solved using a greedy algorithm efficiently. Its ability to handle continuous values sets it apart from the 0/1 Knapsack Problem and makes it suitable for various real-world scenarios.
Input:
Maximum weight = 50. List of items with value and weight.
{(60, 10), (100, 20), (120, 30)}
Output:
Maximum value: 240
By taking the items of weight 20 and 30