To find the optimum solution by exhaustive enumeration, we would need to consider all possible combinations of y vertices out of the rn vertices in the graph. This is a combinatorial problem known as the “Steiner Tree Problem” or “Steiner Tree Enumeration.” The Steiner Tree Problem seeks to find the minimum spanning tree that spans a given subset of vertices, called Steiner points, in addition to the terminal vertices.
The number of minimum spanning tree problems that would need to be solved for exhaustive enumeration can be calculated using the binomial coefficient, also known as “n choose y,” which gives the number of ways to choose y elements from a set of n elements without regard to the order.
The binomial coefficient “n choose y” is calculated as:
C(n, y) = n! / (y! * (n – y)!)
Where n! denotes the factorial of n.
In the context of the problem, we have:
- Total number of vertices: rn
- Number of required vertices in the Steiner tree: y
So, the number of minimum spanning tree problems that would need to be solved is:
C(rn, y) = (m)! / (y! * (m – y)!)
For each combination of y vertices, you would need to solve a minimum spanning tree problem, which involves finding the minimum-weight tree that spans those vertices.
Keep in mind that exhaustive enumeration can quickly become computationally infeasible for larger values of m and y due to the exponential growth of possibilities. In practice, heuristic or approximation algorithms are often used to tackle the Steiner Tree Problem efficiently.