Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum.

```Examples:
set[] = {3, 34, 4, 12, 5, 2},
sum = 9

Output:  True
//There is a subset (4, 5) with sum 9.
```

Let isSubSetSum(int set[], int n, int sum) be the function to find whether there is a subset of set[] with sum equal to sum.

n is the number of elements in set[].

The isSubsetSum problem can be divided into two subproblems :

```Include the last element, recur for n = n-1, sum = sum – set[n-1]
Exclude the last element, recur for n = n-1
```

If any of the above the above subproblems return true, then return true.

# Recursive formula for isSubsetSum() problem

```isSubsetSum(set, n, sum) = isSubsetSum(set, n-1, sum) || isSubsetSum(arr, n-1, sum-set[n-1])

Base Cases
1. isSubsetSum(set, n, sum) = false, if sum > 0 and n == 0
2. isSubsetSum(set, n, sum) = true, if sum == 0```

# Naive recursive implementation that simply follows the recursive structure mentioned above

```// Returns true if there is a subset of set[] with sun equal to given sum
function isSubsetSum(\$set, \$n, \$sum)
{
// Base Cases
if (\$sum == 0)
return true;
if (\$n == 0 && \$sum != 0)
return false;

// If last element is greater than sum, then ignore it
if (\$set[\$n-1] > \$sum)
return isSubsetSum(\$set, \$n-1, \$sum);

/* else, check if sum can be obtained by any of the following
(a) including the last element
(b) excluding the last element   */
return isSubsetSum(\$set, \$n-1, \$sum) || isSubsetSum(\$set, \$n-1, \$sum-\$set[\$n-1]);
}

```
``````// Driver program to test above function
function main(){
\$set = array(3, 34, 4, 12, 5, 2);
\$sum = 9;
\$n = sizeof(\$set)/sizeof(\$set);

if (isSubsetSum(\$set, \$n, \$sum) == true)
print("Found a subset with given sum");
else
print("No subset with given sum");
}

main();``````

Found a subset with given sum

The above solution may try all subsets of given set in worst case. Therefore time complexity of the above solution is exponential. The problem is in-fact NP-Complete (There is no known polynomial time solution for this problem).