Fractional Knapsack Problem

• Difficulty Level : Medium
• Last Updated : 28 Sep, 2021

Given weights and values of n items, we need to put these items in a knapsack of capacity W to get the maximum total value in the knapsack.

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:
Items as (value, weight) pairs
arr[] = {{60, 10}, {100, 20}, {120, 30}}
Knapsack Capacity, W = 50;

Output:
Maximum possible value = 240
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

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

In Fractional Knapsack, we can break items for maximizing the total value of knapsack. This problem in which we can break an item is also called the fractional knapsack problem.

Input :
Same as above

Output :
Maximum possible value = 240
By taking full items of 10 kg, 20 kg and
2/3rd of last item of 30 kg

A brute-force solution would be to try all possible subset with all different fraction but that will be too much time taking.

An efficient solution is to use Greedy approach. The basic idea of the greedy approach is to calculate the ratio value/weight for each item and sort the item on 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.
A simple code with our own comparison function can be written as follows, please see sort function more closely, the third argument to sort function is our comparison function which sorts the item according to value/weight ratio in non-decreasing order.
After sorting we need to loop over these items and add them in our knapsack satisfying above-mentioned criteria.

Below is the implementation of the above idea:

C++

// 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);

//    Uncomment to see new order of Items with their
//    ratio
/*
for (int i = 0; i < n; i++)
{
cout << arr[i].value << "  " << arr[i].weight << " :
"
<< ((double)arr[i].value / arr[i].weight) <<
endl;
}
*/

int curWeight = 0; // Current weight in knapsack
double finalvalue = 0.0; // Result (value in Knapsack)

// Looping through all Items
for (int i = 0; i < n; i++) {
if (curWeight + arr[i].weight <= W) {
curWeight += arr[i].weight;
finalvalue += arr[i].value;
}

// of it
else {
int remain = W - curWeight;
finalvalue += arr[i].value
* ((double)remain
/ (double)arr[i].weight);
break;
}
}

// Returning final value
return finalvalue;
}

// Driver code
int main()
{
int W = 50; //    Weight of knapsack
Item arr[] = { { 60, 10 }, { 100, 20 }, { 120, 30 } };

int n = sizeof(arr) / sizeof(arr);

// Function call
cout << "Maximum value we can obtain = "
<< fractionalKnapsack(W, arr, n);
return 0;
}

Java

// Java program to solve fractional Knapsack Problem
import java.util.Arrays;
import java.util.Comparator;

// Greedy approach
public class FractionalKnapSack {
// function to get maximum value
private static double getMaxValue(int[] wt, int[] val,
int capacity)
{
ItemValue[] iVal = new ItemValue[wt.length];

for (int i = 0; i < wt.length; i++) {
iVal[i] = new ItemValue(wt[i], val[i], i);
}

// sorting items by value;
Arrays.sort(iVal, new Comparator<ItemValue>() {
@Override
public int compare(ItemValue o1, ItemValue o2)
{
return o2.cost.compareTo(o1.cost);
}
});

double totalValue = 0d;

for (ItemValue i : iVal) {

int curWt = (int)i.wt;
int curVal = (int)i.val;

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;
}
}

}

// item value class
static class ItemValue {
Double cost;
double wt, val, ind;

// item value function
public ItemValue(int wt, int val, int ind)
{
this.wt = wt;
this.val = val;
this.ind = ind;
cost = new Double((double)val / (double)wt);
}
}

// Driver code
public static void main(String[] args)
{
int[] wt = { 10, 40, 20, 30 };
int[] val = { 60, 40, 100, 120 };
int capacity = 50;

double maxValue = getMaxValue(wt, val, capacity);

// Function call
System.out.println("Maximum value we can obtain = "
+ maxValue);
}
}

Python3

# Python3 program to solve fractional
# Knapsack Problem

class ItemValue:

"""Item Value DataClass"""

def __init__(self, wt, val, ind):
self.wt = wt
self.val = val
self.ind = ind
self.cost = val // wt

def __lt__(self, other):
return self.cost < other.cost

# Greedy Approach

class FractionalKnapSack:

"""Time Complexity O(n log n)"""
@staticmethod
def getMaxValue(wt, val, capacity):
"""function to get maximum value """
iVal = []
for i in range(len(wt)):
iVal.append(ItemValue(wt[i], val[i], i))

# sorting items by value
iVal.sort(reverse=True)

totalValue = 0
for i in iVal:
curWt = int(i.wt)
curVal = int(i.val)
if capacity - curWt >= 0:
capacity -= curWt
totalValue += curVal
else:
fraction = capacity / curWt
totalValue += curVal * fraction
capacity = int(capacity - (curWt * fraction))
break

# Driver Code
if __name__ == "__main__":
wt = [10, 40, 20, 30]
val = [60, 40, 100, 120]
capacity = 50

# Function call
maxValue = FractionalKnapSack.getMaxValue(wt, val, capacity)
print("Maximum value in Knapsack =", maxValue)

# This code is contributed by vibhu4agarwal

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);
}
}
}

// Driver code
static void Main(string[] args)
{
item[] arr = { new item(60, 10),
new item(100, 20),
new item(120, 30) };

Console.WriteLine("Maximum value we can obtain = " +
FracKnapSack(arr, 50));
}
}

// This code is contributed by Mohamed Adel
Output

Maximum value we can obtain = 240

As main time taking step is sorting, the whole problem can be solved in O(n log n) only.