Can You Solve the Following Minimization Problems?

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I've spent some time on the following problem, which is quite generic:

You are given $$$2n$$$ pairs of integers $$$(x_i, y_i)$$$, where $$$1 \leq x_i, y_i \leq M$$$. Your goal is to choose exactly $$$n$$$ of them, such that their pointwise sum $$$(X, Y)$$$ minimizes the cost function $$$f(X, Y)$$$. What is the best time complexity you can achieve, given some properties about the function $$$f$$$?

Since the time complexity can depend on both $$$n$$$ and $$$M$$$, the order in which the best complexity is determined is:

Polynomial in $$$n$$$ and in $$$\log M$$$.
Polynomial in $$$n$$$ and in $$$M$$$.
$$$O(\binom{2n}{n})$$$ (applicable to all of them).

Try to solve each of the following variations, with varying difficulties. This post is purposed for all levels :).

I'm interested to know if you have any other variations with interesting insights that I can add to here, as I find this topic quite interesting to research.

$$$f(X,Y) = X + Y.$$$

My best time complexity

Solution