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 Hungarian Algorithm: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 04:while ( G_(u,v)) has no perfect matching5: 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 nX8: do u[ i ] A u[ i ] –“9:For each I 2 B \ X10: do v[ i ] A v[ i ] +“11:return perfect matching of ( G_(u,v))

## Example

Three workers to do Three task at the same time. Apply the Hungarian algorithm the cost matrix is,

### Solution:

STEP 1:

Identify minimum in each row (show in ‘green’)

 TASK_A TASK_B TASK_C ROW_MINS WORKER_1 250 450 350 250 WORKER_2 400 400 350 350 WORKER_3 200 500 250 200

Substract row mins from each element in their respective rows:

 TASK_A TASK_B TASK_C WORKER_1 0 200 100 WORKER_2 50 50 0 WORKER_3 200 300 50 COL_MINS 0 50 0

Identify minimum in each column (show in ‘green’).

Substract column mins from each element from their respective columns:

This is reduced cost matrix.

STEP 2:

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:

STEP 3:

Find the minimum number of horizontal and vertical lines to cover all 0’s: