Question

Every year, Professor Dumbledore assigns the instructors at Hogwarts to various faculty committees.

There are n faculty members and c committees. Each committee member has submitted a list of
their prices for serving on each committee; each price could be positive, negative, zero, or even
infinite. For example, Professor Snape might declare that he would serve on the Student Recruiting
Committee for 1000 Galleons, that he would pay 10000 Galleons to serve on the Defense Against
the Dark Arts Course Revision Committee, and that he would not serve on the Muggle Relations
committee for any price.
Conversely, Dumbledore knows how many instructors are needed for each committee, as
well as a list of instructors who would be suitable members for each committee. (For example:
"Dark Arts Revision: 5 members, anyone but Snape.") If Dumbledore assigns an instructor to a
committee, he must pay that instructor’s price from the Hogwarts treasury.
Dumbledore needs to assign instructors to committees so that (1) each committee is full, (3) no
instructor is assigned to more than three committees, (2) only suitable and willing instructors
are assigned to each committee, and (4) the total cost of the assignment is as small as possible.
Describe and analyze an efficient algorithm that either solves Dumbledore’s problem, or correctly
reports that there is no valid assignment whose total cost is finite

Answer 1

The Hungarian Algorithm is an efficient algorithm that can solve Dum`bledore's problem of assigning instructors to committees while minimizing the total cost. It involves creating a matrix of costs, subtracting row and column minimums, finding and covering zeros, and updating the matrix until an optimal assignment is achieved.

Step-by-step explanation:

To solve Dum`bledore's problem efficiently, we can use the concept of a min-cost max-flow algorithm. This algorithm combines the concepts of flow networks and minimum-cost matching to find the optimal assignment of instructors to committees while minimizing the total cost.

Here's an overview of the algorithm:

1. Create a flow network representation of the problem. Each instructor will be represented by a node, and each committee will be represented by a node. Add a source node and a sink node.

2. Add edges between the source node and the instructors. The capacity of these edges will be set to 3, representing the maximum number of committees an instructor can be assigned to. The cost of these edges will be 0.

3. Add edges between the instructors and the committees they are suitable for. The capacity of these edges will be set to 1, representing the number of instructors needed for each committee. The cost of these edges will be the price declared by the instructor for serving on that committee.

4. Add edges between the committees and the sink node. The capacity of these edges will be set to the number of instructors needed for each committee. The cost of these edges will be 0.

5. Run the min-cost max-flow algorithm on the flow network. This algorithm will find the maximum flow from the source to the sink while minimizing the total cost.

6. Once the algorithm is complete, the assignments can be determined based on the flow. If an edge from an instructor to a committee has flow equal to 1, it means that the instructor is assigned to that committee. The total cost of the assignment is equal to the minimum cost found by the algorithm.

7. If there is no valid assignment whose total cost is finite, the algorithm will report it by detecting that the total flow from the source to the sink is less than the total number of instructors.

By using the min-cost max-flow algorithm, we can efficiently solve Dum`bledore's problem and find the optimal assignment of instructors to committees while minimizing the total cost. The algorithm guarantees that all the given constraints are satisfied, such as committee fullness, instructor limits, suitability, and willingness.

Answer 2

Step-by-step explanation:

Base on the scenario been describe in the question, the algorithm that describe professor Dumbledore’s problem, or correctly

reports that there is no valid assignment whose total cost is finite is written as follows; Dumbledore needs to assign instructors to committees so that (1) each committee is full, (3) no

instructor is assigned to more than three committees, (2) only suitable and willing instructors

are assigned to each committee, and (4) the total cost of the assignment is as small as possible.

Describe and analyze an efficient algorithm that either solves Dumbledore’s problem, or correctly

reports that there is no valid assignment whose total cost is finite

.