A common problem in clinical studies is how to choose age-, and usually also gender-, matched pairs from two groups such as case/control. (case=has disease, control=not have).
After searching the web a bit, it seems like many researchers do it sort of by hand. To me it seems like this might fall into some standard class of optimization problems. I also haven't found an algorithm that tackles it from that point of view. The objective function could be something like the root-mean-square age difference.
Suppose I have $N$ subjects in the case group and $M$ subjects in the control group, with $N>M$. (In my case, $N=85, M=49$). Without doing any kind of sorting or pre-selection, i.e. being completely naive, there are an enormous number of potential pairings. Here is my count (I hope I'm doing this right):
Take the $M$ controls and put them in an arbitrary order. The first of these can be associated with a random one of $N$ in the case group. The next, with $N-1$, and so on down the line. So the total number of pairings is:
$$N(N-1)(N-2)\cdots(N-M+1) = \frac{N!}{(N-M)!}$$
That's a huge number. But maybe smart math people have figured out a clever solution to this problem.
Is this the best forum for this question?
Here is an example way to fill in a specific model:
You have $m$ controls and $n$ subjects with $m \leq n$. You have ages $a_1,...,a_m$ and $b_1,...,b_n$. Define
$$w_{ij}=(a_i-b_j)^2$$
You want to find a binary matching matrix $x=(x_{ij})$ that solves the “min weight match” problem of minimizing $$ \sum_{i =1}^m\sum_{j=1}^n x_{ij}w_{ij}$$ subject to $(x_{ij})$ being a valid matching. This is polynomially solvable and is equivalent to “max weight matching” under a simple transformation of weights.
I should mention the problem with general weights $w_{ij}$ is polynomial solvable via certain methods, but in fact the quadratic weight problem may be even easier...