Given a matrix of size N X N, the task is to find maximum sum of this Matrix where each value picked is from a unique column for every row.
Examples:
Input: matrix = [[3, 4, 4, 4], [1, 3, 4, 4], [3, 2, 3, 4], [4, 4, 4, 4]] Output: 16 Explanation: Selecting (0, 1) from row 1 = 4 Selecting (1, 2) from row 2 = 4 Selecting (2, 3) from row 3 = 4 Selecting (3, 0) from row 4 = 4 Therefore, max sum = 4 + 4 + 4 + 4 = 16 Input: matrix = [[0, 1, 0, 1], [3, 0, 0, 2], [1, 0, 2, 0], [0, 2, 0, 0]] Output: 8 Explanation: Selecting (0, 3) from row 1 = 1 Selecting (1, 0) from row 2 = 3 Selecting (2, 2) from row 3 = 2 Selecting (3, 1) from row 4 = 2 Therefore, max sum = 1 + 3 + 2 + 2 = 8
Approach:
 Genrate a numeric string of size N containing numbers from 1 to N
 Find the permutation of this string (N!).
 Now pairing is done between the permutations, such that each N! pairing has a unique column for every row.
 Then calculate the sum of values for all the pairs.
Below is the implementation of the above approach:
# Python code for maximum sum of # a Matrix where each value is # from a unique row and column # Permutations using library function from itertools import permutations
# Function MaxSum to find # maximum sum in matrix def MaxSum(side, matrix):
s = ''
# Generating the string
for i in range ( 0 , side):
s = s + str (i)
# Permutations of s string
permlist = permutations(s)
# List all possible case
cases = []
# Append all possible case in cases list
for perm in list (permlist):
cases.append(''.join(perm))
# List to store all Sums
sum = []
# Iterate over all case
for c in cases:
p = []
tot = 0
for i in range ( 0 , side):
p.append(matrix[ int (s[i])][ int (c[i])])
p.sort()
for i in range (side  1 ,  1 ,  1 ):
tot = tot + p[i]
sum .append(tot)
# Maximum of sum list is
# the max sum
print ( max ( sum ))
# Driver code if __name__ = = '__main__' :
side = 4
matrix = [[ 3 , 4 , 4 , 4 ], [ 1 , 3 , 4 , 4 ], [ 3 , 2 , 3 , 4 ], [ 4 , 4 , 4 , 4 ]]
MaxSum(side, matrix)
side = 3
matrix = [[ 1 , 2 , 3 ], [ 6 , 5 , 4 ], [ 7 , 9 , 8 ]]
MaxSum(side, matrix)

16 18
Time complexity: O(K), where K = N!
Auxiliary Space complexity: O(K), where K = N!
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