Problems with minimum and maximum variants and non-uniform samples?

Question

Algorithms

Please give links to the material on how to solve such patterns (theory):

There are n boards, you need to build a fence of m boards, while the fence should be as high as possible (the boards can be sawn as a whole (that is, int)), each i board has Xi height.

That is, at the entrance
8 - how many boards should be in the fence, 4 - how many there are
x1, x2, x3, x4 - the boards themselves (or rather their heights)

The tasks seem interesting, so I want to tighten up the theory

Answer 1

Such problems are often solved by dichotomy + greedy agoritm, as in this example (you need to sort through the answer by binary search, and then see if you can get the required number of boards of a given length from the original ones).
I can't think of any general theory. Still, with discrete optimization, there are a lot of different problems. I can only advise you to look at greedy algorithms. In addition to solutions, be sure to learn how to prove solutions and why greed works in each case.