1. HUNGARIAN ALGORITHM

2. Abstract
Unfinished business from where many participants did not get to experience all the depth of questions possible with the Hungarian Algorithm. This session will be organised in the form of a workshop in which teachers can work collaboratively to use the algorithm for assignment/allocation applications. Prior knowledge of the algorithm is not required.

3. Syllabus reference
Assignment problems use a bipartite graph and/or its tabular or matrix form to represent an assignment/ allocation problem determine the optimum assignment(s), by inspection for small-scale problems, or by use of the Hungarian algorithm for larger problems Glossary: The Hungarian algorithm is used to solve assignment (allocation) problems.

4. ACTIVITIES WACE 2016 exam question
For beginners – The Simpsons at work
Multiple steps and more than 1 solution
Maximising
Outings for the elderly
An engineering competition
More tasks than people
5 by 5 matrix
Investigate – making more matrices
Writing interesting (?) questions

5. A1: WACE exam : 2016 Complete the matrix for the bipartite graph

6. WACE exam : 2016 Complete the matrix for the bipartite graph4 7 8 2 6 3 5 9
From every number in each row subtract the smallest number in that row

7. WACE exam : 2016 Subtracting the smallest number in each row3 4 1 2 5
From every number in each column subtract the smallest number in that column

8. WACE exam : 2016 Subtracting the smallest number in each column1 3 4 2 5
Using vertical or horizontal lines across rows or down columns, cross out all zeroes using as few lines as possible

9. WACE exam : 2016 Crossing out zeroes using as few lines as possible1 3 4 2 5
Note: the number of lines is less than the dimensions of the matrix so a solution does not exist yet.

10. WACE exam : 2016 Subtract the lowest uncovered number from all uncovered numbers and add it to the numbers at the intersections.
1 0
3 2
4 3
2 1
1 2 3
5 6 1 2
Using vertical or horizontal lines across rows or down columns, cross out all zeroes using as few lines as possible

11. WACE exam : 2016 Cross out zeroes using horizontal or vertical lines across rows or down columns – use as few lines as possible 2 3 1 6
Note: the number of lines is equal to the dimensions of the matrix so a solution exists.

12. WACE exam : 2016 To assign – locate row or column with only 1 zero
WACE exam : 2016 To assign – locate row or column with only 1 zero. Allocate ... that person or job is no longer available. Continue until all jobs assigned. 2 3 1 6
Row 4 column 3 OR Row 3 column 4 then the other.
Then row 2 column 1 and then row 1 column 2.

13. Assignment to minimise total time
In each row take the smallest number off every number in that row
In each column take the smallest number off every number in that column
** Using the least number of vertical or horizontal lines across columns or rows, cover all zeroes
If the number of lines equals the dimensions of the matrix a solution is “found” – Go to END – otherwise continue
Locate smallest number not crossed out
Take this number off every uncovered number and add it to numbers at the intersections of the lines
“Remove” vertical and horizontal lines
Back to **
END: Locate a zero in each row and column so that there is “one for each”
Determine the total minimum time for all tasks.

14. ACTIVITIES WACE 2016 exam question 
For beginners – The Simpsons at work
Multiple steps and more than 1 solution
Maximising
Outings for the elderly
An engineering competition
More tasks than people
5 by 5 matrix
Investigate – making more matrices

15. A2: The Simpsons at work Minimising total time
In each row take the smallest number off every number in that row
In each column take the smallest number off every number in that column
Using as few vertical or horizontal lines as possible, remove all zeroes
If no. lines = matrix dimensions, solution exists
Locate a zero in each row and column so that there is “one for each”
Determine the total minimum time for all 4 tasks. Can a bipartite graph be used to represent the solution?

16. A3: Event Managers Minimising total time
In each row subtract the smallest number from every number in that row
In each column subtract the smallest number from every number in that column
** Using as few vertical or horizontal lines as possible, cover all zeroes
If the number of lines equals the dimensions of the matrix a solution is “found” – Go to END – otherwise continue
Locate smallest number not crossed out
Take this number off every uncovered number and add it to numbers at the intersections of the lines
Back to **
END: Locate a zero in each row and column so that there is “one for each”
Determine the total minimum time for all 4 bulldozers. How can a bipartite graph be used to represent the solution?

17. A4: Maximising
Firstly – take every number in the matrix off the largest number and then proceed as before.
Council outings for the elderly
Mathematics competition for students

18. A5: Cleaning up the Art room
How is this situation different to the ones presented earlier?

19. Cleaning up the Art room
There are more people than tasks
Number of supply sources exceeds number of demand depots
The matrix formed is not square
We make it square by adding a row or column
By convention the row or column added is filled with the largest number in the matrix
Proceed as before

20. A6: 5 x 5

21. A7: Investigation Writing questions on the Hungarian algorithm
The context
The numbers
How can you make up a set of numbers for the HA manipulating a matrix that already works in the desired manner?

22. Make a conjecture. Test it with two matrices.
Group 1
Add the same number to every number in the matrix
Group 2
Add a different number to the elements in each row
Group 3
Multiply all the numbers in the matrix by the same number
Group 4
Multiply every number in each row by a different number for each row
Make a conjecture. Test it with two matrices.
What can you conclude? Share your findings.

23. Further resources https://www.youtube.com/watch?v=cQ5MsiGa DY8PM
mathematics/further/exams.aspx
See paper 2 – 2012, 2014
MAWA – IQ Topic 4.3