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Weighed matching (k_(n,n)= A[B, A Є B]), w[n, n])
1:for each i 2 [ n ]
2: do u[ i ] A maxj w[ (i, j )]
3: v[ i ]A 0
4:while ( G_(u,v)) has no perfect matching
5: do X A minimum vertex cover of ( G_(u,v))
6:“ A min u[ i ] + v[ j ] - w[ i, j ] : fi, jg2 (A n X Є (B n X))
7:for each I 2 A nX
8: do u[ i ] A u[ i ] –“
9:For each I 2 B \ X
10: do v[ i ] A v[ i ] +“
11:return perfect matching of ( G_(u,v))
Three workers to do Three task at the same time. Apply the Hungarian algorithm the cost matrix is,
Identify minimum in each row (show in ‘green’)
Substract row mins from each element in their respective rows:
Identify minimum in each column (show in ‘green’).
Substract column mins from each element from their respective columns:
This is reduced cost matrix.
Find the minimum no of horizontal and vertical lines to cover all 0’s:
Since < 3 lines suffice, continue to step2.
Identify min of uncovered elements: uncovered min = 50
Substract min of uncovered matrix elements from the uncovered rows:
Add the min of the uncovered matrix elements to the covered columns:
Find the minimum number of horizontal and vertical lines to cover all 0’s:
Since it takes 3 linkes to cover all zero’s, STOP: an assignment is possible.
Construct a valid assignment of row to column labels using only matrix zeros: