Fractional Knapsack | Algorithms - Computer Science Engineering (CSE) PDF Download

Introduction

Given the weights and values of N items, in the form of {value, weight} put these items in a knapsack of capacity W to get the maximum total value in the knapsack. In Fractional Knapsack, we can break items for maximizing the total value of the knapsack
Note: In the 0-1 Knapsack problem, we are not allowed to break items. We either take the whole item or don’t take it.

Input: arr[] = {{60, 10}, {100, 20}, {120, 30}}, W = 50

Output: 240 

Explanation: By taking items of weight 10 and 20 kg and 2/3 fraction of 30 kg. 

Hence total price will be 60+100+(2/3)(120) = 240

Input:  arr[] = {{500, 30}}, W = 10

Output: 166.667

Naive Approach: To solve the problem follow the below idea:

Try all possible subsets with all different fractions.

Fractional Knapsack Problem using Greedy algorithm

An efficient solution is to use the Greedy approach. 

The basic idea of the greedy approach is to calculate the ratio value/weight for each item and sort the item on the basis of this ratio. Then take the item with the highest ratio and add them until we can’t add the next item as a whole and at the end add the next item as much as we can. Which will always be the optimal solution to this problem.

Follow the given steps to solve the problem using the above approach:

• Calculate the ratio(value/weight) for each item.
• Sort all the items in decreasing order of the ratio.
• Initialize res =0, curr_cap = given_cap.
• Do the following for every item “i” in the sorted order:
  • If the weight of the current item is less than or equal to the remaining capacity then add the value of that item into the result
  • Else add the current item as much as we can and break out of the loop.
• Return res.

Below is the implementation of the above approach:

C++

// C++ program to solve fractional Knapsack Problem

 

#include <bits/stdc++.h>

using namespace std;

 

// Structure for an item which stores weight and

// corresponding value of Item

struct Item {

    int value, weight;

 

    // Constructor

    Item(int value, int weight)

    {

        this->value = value;

        this->weight = weight;

    }

};

 

// Comparison function to sort Item

// according to val/weight ratio

bool cmp(struct Item a, struct Item b)

{

    double r1 = (double)a.value / (double)a.weight;

    double r2 = (double)b.value / (double)b.weight;

    return r1 > r2;

}

 

// Main greedy function to solve problem

double fractionalKnapsack(int W, struct Item arr[], int N)

{

    // Sorting Item on basis of ratio

    sort(arr, arr + N, cmp);

 

    double finalvalue = 0.0;

 

    // Looping through all items

    for (int i = 0; i < N; i++) {

         

        // If adding Item won't overflow,

        // add it completely

        if (arr[i].weight <= W) {

            W -= arr[i].weight;

            finalvalue += arr[i].value;

        }

 

        // If we can't add current Item,

        // add fractional part of it

        else {

            finalvalue

                += arr[i].value

                   * ((double)W / (double)arr[i].weight);

            break;

        }

    }

 

    // Returning final value

    return finalvalue;

}

 

// Driver code

int main()

{

    int W = 50;

    Item arr[] = { { 60, 10 }, { 100, 20 }, { 120, 30 } };

 

    int N = sizeof(arr) / sizeof(arr[0]);

 

    // Function call

    cout << fractionalKnapsack(W, arr, N);

    return 0;

}

Java

// Java program to solve fractional Knapsack Problem

 

import java.io.*;

import java.util.Arrays;

import java.util.Comparator;

 

// Greedy approach

class FractionalKnapSack {

     

    // Function to get maximum value

    private static double getMaxValue(ItemValue[] arr,

                                      int capacity)

    {

        // Sorting items by value/weight ratio;

        Arrays.sort(arr, new Comparator<ItemValue>() {

            @Override

            public int compare(ItemValue item1,

                               ItemValue item2)

            {

                double cpr1

                    = new Double((double)item1.value

                                 / (double)item1.weight);

                double cpr2

                    = new Double((double)item2.value

                                 / (double)item2.weight);

 

                if (cpr1 < cpr2)

                    return 1;

                else

                    return -1;

            }

        });

 

        double totalValue = 0d;

 

        for (ItemValue i : arr) {

 

            int curWt = (int)i.weight;

            int curVal = (int)i.value;

 

            if (capacity - curWt >= 0) {

 

                // this weight can be picked while

                capacity = capacity - curWt;

                totalValue += curVal;

            }

            else {

 

                // Item cant be picked whole

                double fraction

                    = ((double)capacity / (double)curWt);

                totalValue += (curVal * fraction);

                capacity

                    = (int)(capacity - (curWt * fraction));

                break;

            }

        }

 

        return totalValue;

    }

 

    // Item value class

    static class ItemValue {

 

        int value, weight;

 

        // Item value function

        public ItemValue(int val, int wt)

        {

            this.weight = wt;

            this.value = val;

        }

    }

 

    // Driver code

    public static void main(String[] args)

    {

 

        ItemValue[] arr = { new ItemValue(60, 10),

                            new ItemValue(100, 20),

                            new ItemValue(120, 30) };

 

        int capacity = 50;

 

        double maxValue = getMaxValue(arr, capacity);

 

        // Function call

        System.out.println(maxValue);

    }

}

Python3

# Structure for an item which stores weight and

# corresponding value of Item

class Item:

    def __init__(self, value, weight):

        self.value = value

        self.weight = weight

 

# Main greedy function to solve problem

def fractionalKnapsack(W, arr):

 

    # Sorting Item on basis of ratio

    arr.sort(key=lambda x: (x.value/x.weight), reverse=True)   

 

    # Result(value in Knapsack)

    finalvalue = 0.0

 

    # Looping through all Items

    for item in arr:

 

        # If adding Item won't overflow,

        # add it completely

        if item.weight <= W:

            W -= item.weight

            finalvalue += item.value

 

        # If we can't add current Item,

        # add fractional part of it

        else:

            finalvalue += item.value * W / item.weight

            break

     

    # Returning final value

    return finalvalue

 

 

# Driver Code

if __name__ == "__main__":

 

    W = 50

    arr = [Item(60, 10), Item(100, 20), Item(120, 30)]

 

    # Function call

    max_val = fractionalKnapsack(W, arr)

    print(max_val)

C#

// C# program to solve fractional Knapsack Problem

 

using System;

using System.Collections;

 

class GFG {

 

    // Class for an item which stores weight and

    // corresponding value of Item

    class item {

        public int value;

        public int weight;

 

        public item(int value, int weight)

        {

            this.value = value;

            this.weight = weight;

        }

    }

 

    // Comparison function to sort Item according

    // to val/weight ratio

    class cprCompare : IComparer {

        public int Compare(Object x, Object y)

        {

            item item1 = (item)x;

            item item2 = (item)y;

            double cpr1 = (double)item1.value

                          / (double)item1.weight;

            double cpr2 = (double)item2.value

                          / (double)item2.weight;

 

            if (cpr1 < cpr2)

                return 1;

 

            return cpr1 > cpr2 ? -1 : 0;

        }

    }

 

    // Main greedy function to solve problem

    static double FracKnapSack(item[] items, int w)

    {

 

        // Sort items based on cost per units

        cprCompare cmp = new cprCompare();

        Array.Sort(items, cmp);

 

        // Traverse items, if it can fit,

        // take it all, else take fraction

        double totalVal = 0f;

        int currW = 0;

 

        foreach(item i in items)

        {

            float remaining = w - currW;

 

            // If the whole item can be

            // taken, take it

            if (i.weight <= remaining) {

                totalVal += (double)i.value;

                currW += i.weight;

            }

 

            // dd fraction until we run out of space

            else {

                if (remaining == 0)

                    break;

 

                double fraction

                    = remaining / (double)i.weight;

                totalVal += fraction * (double)i.value;

                currW += (int)(fraction * (double)i.weight);

            }

        }

        return totalVal;

    }

 

    // Driver code

    static void Main(string[] args)

    {

        item[] arr = { new item(60, 10), new item(100, 20),

                       new item(120, 30) };

 

        // Function call

        Console.WriteLine(FracKnapSack(arr, 50));

    }

}

 

// This code is contributed by Mohamed Adel

Output

Maximum value we can obtain = 240

Time Complexity: O(N * log N)
Auxiliary Space: O(N)