Calculating the highest 2-side average match - excel

Question

"First part of the question is dedicated towards explaining the concept better, so we know, what we're calculating with. Feel free to
skip below to the latter parts, if you find it unnecessary"
1. Basic overview of the question:
Hello, I've got an excel application, something akin to a dating site. You can open various user profiles and even scan through the data and find the potential matches, based on hobbies, cities and other criteria.
How it's calculated is not relevant to the question, but the result of
a "Find Match" calculation looks something like this, a sorted list
of users, depending on how fitting they are (last column)
Relevant to the question are mainly:
the first column (ID) - ID of the user
the last column (Zhoda) - Match% of other users, against the one currently selected
2. What I need to do - how it's currently done
I need to find the highest match on average out of all users. If I were to write this algorhitmically:
1. Loop through all users
2. For each user in our database calculate the potential matches
3. Store the score of selected user ID, against all the found user IDs
4. Once it's all calculated, pit all users against each other _
and find the highest match on average
Obviously that sounds pretty complicated / vague, so here's a
simplified example. Let's say I have completed the first 3 steps and
have gotten the following result:
Here, the desired result would be:
User1 <- 46% -> User2
as they have the highest combined percentage average:
User1 vs User2: 30%
User2 vs User1: 62%
User1 <- (30+62)/2 -> User2
And no other possible combination of users has higher match% average
3. The purpose behind the question:
Now obviously you may ask, if I get the calculation behind it, then why ask the question in the first place? Well, the reason is combination of everything vs everything is extremely inefficient.
As soon as there are let's say 100 users instead of 3 in my database. I would have to do 100*100 calculations on match% alone, let alone afterwards check the average Match% of each individual user against another.
Is there perhaps some better way to approach, in a way I could either
minimize the data I have to calculate with
some sorting algorithm, where I could skip certain calculations in order to be quicker
an overall better approach towards calculating the highest average match%
So to recapitulate:
I've got a database of users.
Each individual user has a certain amount of Match% against every other user
I need to find two users, who one against another (on both sides) have the highest Match% average out of all possible combinations.
If you feel like you need any additional info, please let me know.
I'll try to keep the question updated as much as possible.

Answer 1

As you've presented the problem -- no, you cannot speed this up significantly. Since you've presented match% as an arbitrary function, constrained only by implied range, there are no mathematical properties you can harness to reduce the worst-case search scenario.
Under the given circumstances, the best you can do is to leverage the range. First, don't bother with "average": since these are strictly binary matches, dividing by 2 is simply a waste of time; keep the total.
Start by picking a pair; do the two-way match. Once you find a total of more than 100, store that value and use it to prune any sub-standard searches. For instance, if your best match so far totals 120, then if you find a couple where match(A, B) < 20, you don't bother with computing match(B, A).
In between, you can maintain a sorted list (O(n log n)) of first matches; don't do the second match unless you have reason to believe that this one might exceed your best match.
The rest of your optimization consists of gathering statistics about your matching, so that you can balance when to do first-only against two-way matches. For instance, you might defer the second match for any first match that is below the 70th percentile of those already deferred. This is in hope of finding a far better match that would entire eliminate this one.
If you gather statistics on the distribution of your match function, then you can tune this back-and-forth process better.
If you can derive mathematical properties about your match function, then there may be ways to leverage those properties for greater efficiency. However, since it's already short of being a formal topological "distance" metric d (see below), I don't hold out much hope for that.
Basic metric properties:
d(A, B) exists for all pairs (A, B)
d(A, B) = d(B, A)
d(A, A) = 0 // does not apply to a bipartite graph
d obeys the triangle inequality -- which doesn't apply directly, but has some indirect consequences for a bipartite graph.