1. Review: Mini-Quiz Matching
Matching Homework 1 2 3 4 5 6 A X B C D E F
a) Show all possible allocations on a bipartite graph.
b) Use an appropriate algorithm to find a maximal matching on a bipartite graph.
c) List any alternate paths you have used. (Show your work: graph 1, graph 2, etc. )
d) Explain why it is not possible to find a complete matching.

2. Class Greeting

3. Objective: The student will be able to solve Cross Topic Mathematical Problems.

4. You are choosing your courses…
You want to study Math, Physics, French and Philosophy
You also want to attend Basketball Club
Each of these is available in one or more of 6 different option blocks
Subject
Option Blocks
Maths
A, B, F
Physics
D, E, F
French
A, B
Philosophy
C, E
Basketball Club C

6. if your combination of courses is possible?
How do you decide
if your combination of courses is possible?

7. Let’s summarise the problem
We have a set of subjects and a set of option blocks with each subject available in one or more of the option blocks.

8. Draw a bipartite graph to represent the availability of subjects.
Explain why it is not possible to find a complete matching.
The student would like to avoid having lessons in Option E because this is taught on a Friday afternoon. Explain why this is not possible.
Having accepted that he cannot avoid Option E, the student would like to try to avoid Option A, which is taught on a Monday morning. Find a maximal matching that does not include Option A.
The student suddenly realises that the College also offers guitar lessons, available only in Option D. Even though it will mean that he has a full timetable (and has to get out of bed on a Monday morning) he would like to also attend these. With your maximal matching from (d) as a starting point, use the Alternating Path Algorithm to find a complete matching

9. Solution:
Draw a bipartite graph to represent the availability of subjects.
a) The bipartite graph is shown below.
MATHS
PHYSICS
FRENCH
PHILOSOPHY
BASKETBALL A B C D E F
It is only possible to find a complete matching when each set has the same number of nodes.
Explain why it is not possible to find a complete matching.

10. The student would like to avoid having lessons in Option E because this is taught on a Friday afternoon. Explain why this is not possible.
MATHS
PHYSICS
FRENCH
PHILOSOPHY
BASKETBALL A B C D E F
Basketball Club is only available in Option C and Philosophy only in C and E. The student will therefore have to do Basketball in C and Philosophy in E.   

11. Having accepted that he cannot avoid Option E, the student would like to try to avoid Option A, which is taught on a Monday morning. Find a maximal matching that does not include Option A.
MATHS
PHYSICS
FRENCH
PHILOSOPHY
BASKETBALL A B C D E F
A maximal matching is
  Maths - F
Physics – D
French – B
Philosophy – E
Basketball – C

12. An alternating path is: Guitar – D = Physics – F = Maths – A
The student suddenly realises that the College also offers Guitar Lessons, available only in Option D. Even though it will mean that he has a full timetable (and has to get out of bed on a Monday morning) he would like to also attend these. With your maximal matching from (d) as a starting point, use the Alternating Path Algorithm to find a complete matching
MATHS
PHYSICS
FRENCH
PHILOSOPHY
BASKETBALL A B C D E F
GUITAR
An alternating path is:
Guitar – D = Physics – F = Maths – A
This gives a complete matching of:
  Maths – A
Physics – F
French – B
Philosophy – E
Basketball – C
Guitar – D

13. Another example where this is used
British Airways scheduling staff:
deciding who will work, when, and where…

14. Another example where this is used
British Airways scheduling staff:
deciding who will work, when, and where…

15. Another example where this is used
British Airways scheduling staff:
deciding who will work, when, and where…

16. Another example where this is used
British Airways scheduling staff:
deciding who will work, when, and where…

17. Another example where this is used
British Airways scheduling staff:
deciding who will work, when, and where…

18. We want to match up our chosen subjects with possible option blocks in a systematic method.

19. This is an example of a Matching Problem

20. Matching Problems
We have two distinct sets of vertices with some edges between the two sets but no edges within a set

21. We want to find as large as possible a subset of the edges with no two edges sharing a vertex.

22. Problem:
A student at the beginning of year M6 is choosing his course options. He wants to study Math, Physics, French and Philosophy, and he also wants to attend Basketball Club. The College timetable is divided into 6 option blocks labelled A to F. The table below shows which of the student’s chosen courses is available in each option block.
Subject
Option Blocks
Maths
A, B, F
Physics
D, E, F
French
A, B
Philosophy
C, E
Basketball Club C

23. Lesson Summary:
Objective: The student will be able to solve Cross Topic Mathematical Problems.

24. Preview of the Next Lesson:
Objective: The student will be able to solve Mathematical Problems involving estimation techniques.

25. Mathematical Skills and Processes - HW 3
Homework
Mathematical Skills and Processes - HW 3