algorithm for dividing x amount of people into n rooms of different sizes - algorithm

Question

For a project I have to design an algorithm that will fit a group of people into hotel rooms given their preference. I have created a dictionary in Python that has a person as key, and as a value a list of all people they would like to be in a room with.
There are different types of rooms that can hold between 2-10 people. How many rooms of what type there are is specified by the user of the program.
I have tried to brute force this problem by trying all room combinations and then giving each room a score based on the preference of the residents and looking for the maximum score. This works fine for small group sizes but having a group of 200 will give 200! combinations which my poor computer will not be able to compute within my lifetime.
I was wondering if there is an algorithm that I have not been able to find with the solution to my problem.
Thanks in advance!
Thijs

Answer 1

What you can do is think of your dictionary as a graph. Then you can create an adjacency matrix.
For example let say you have a group of 4 people, A, B, C and D.
A: wants to be with B and C
B: wants to be with A
C: wants to be with D
D: want to be with A and C
Your matrix would look like this:
// A B C D
// A 0 1 1 0
// B 1 0 0 0
// C 0 0 0 1
// D 1 0 1 0
Let's call this matrix M. You can then calculate the transpose (let's call it MT) and add M to MT. You will get something like this.
// A B C D
// A 0 2 1 1
// B 2 0 0 0
// C 1 0 0 2
// D 1 0 2 0
Then order the lines (or the columns it doesn't matter because it is symmetric) based on the sum of its values.
// A B C D
// A 0 2 1 1
// C 1 0 0 2
// D 1 0 2 0
// B 2 0 0 0
Do the same with the columns
// A C D B
// A 0 1 1 2
// C 1 0 2 0
// D 1 2 0 0
// B 2 0 0 0
Start filling your rooms starting from the first line based on the greatest value in that line and reduce the matrix by removing people that were assigned a room. You should start by selecting the biggest room first.
For example if we have a room that can have 2 people you'd assign person B and A to it since the biggest value in the first line is 2 and it corresponds to person B.
The reduced matrix would then be:
// C D
// C 0 2
// D 2 0
And you loop till all is done.

Answer 2

You already had a greedy solution described. So instead I'll suggest a simulated annealing solution.
For this you first assign everyone to rooms randomly. And now you start considering swapping people at random. You always accept swaps that improve your score, but have a chance of accepting a bad swap. The chance of accepting a bad swap goes down if the swap is really bad, and also goes down with time. After you've experimented enough, whatever you have is probably pretty good.
It is called "simulated annealing" because it is a simulation of the process by which a slowly cooling substance forms a well-organized crystal structure. So the parameter that you usually use is called T for temperature. And a standard function is:
def maybe_swap(assignment, x, y, T):
score_now = score(assignment)
swapped = swap(assignment, x, y)
score_swapped = score(swapped)
if random.random() < math.exp( (score_swapped - score_now) / T ):
return swapped
else:
return assignment
And then you just have to play around with how much work to do. Something like this:
for count_down in range(400, -1, -1):
for i in range(n^2):
x = floor(random.random(n))
y = floor(random.random(n))
if x != y:
assignment = maybe_swap(assignment, x, y, count_down / 100.0)
(You should play around with the parameters.)