Which polynomial time algorithm would you use to solve this if there is one to solve?

Question

A group of m primary students is given a list of n foods offered by the cafeteria. Each student is asked to write down their 2 favorite foods. Each student must write exactly two food preferences. The cafeteria would like to select at most k students to represent the school at the food fair. The students must be chosen in such a way that each of the n foods is a selected preference of at least one of the k students.

Using the same input as above, suppose now we wish to select at most k food options to put on the school menu in such a way that every child has at least one of their preferences on the menu.

Answer 1

To solve this problem, you can use the Greedy Algorithm. The Greedy Algorithm works by making a locally optimal choice at each step in order to reach a global optimum.

In this particular problem, the algorithm would work by selecting the food option with the most number of student preferences at each step. This would ensure that the food options chosen maximise the number of student preferences and thus, minimise the number of students whose preferences are not included in the final selection.

The algorithm can be represented by the following pseudo-code:

// Inputs: m (number of students), n (number of food options), k (number of food options to select)

// Initialise an empty selection

selection = []

// Iterate k times

for i = 0 to k

// Find the food option with the most number of student preferences

max_pref = 0

max_idx = -1

for j = 0 to n

pref = count_preferences(j)

if pref > max_pref

max_pref = pref

max_idx = j

// Add this food option to the selection

selection.add(max_idx)

// Output the selection

return selection