**Mathematics 243 ** (Combinatorial and Discrete Geometry)

Review: Examination II (Part II)

prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, NY 11451

email: joeyc@cunyvm.cuny.edu

web page: http://www.york.cuny.edu/~malk

1. Use Konig's alternating path algorithm to find a larger matching for the graph below, starting with the partial matching indicated using bold edges:

2. Use Konig's algorithm to find a maximum cardinality matching for the graphs below:

3. Using the matrix of jobs (rows) and workers (columns) below find:

a. A matching whose total weight is as large as possible

b. A matching whose total weight is as small as possible

c. A matching whose maximum weight edge is as small as possible

d. A matching whose smallest weight edge is as large as possible.

Worker 1 | Worker 2 | Worker 3 | |

Job 1 | 16 | 15 | 19 |

Job 2 | 24 | 21 | 22 |

Job 3 | 27 | 25 | 31 |

e. What interpretation of the numbers above might lead one to want to minimize the total weight of a matching?

f. What interpretation of the numbers above might lead one to want to maximize the the total weight of a matching?