In mathematics, in particular in combinatorics, these words have a very specific contrastive usage as adjectives.

• A minimal solution to a problem can't be made any smaller by "shrinking" it. If we shrink it, it is no longer a solution.

• A minimum solution to the problem has the smallest possible size among all solutions. No smaller solutions exist.

This is a very specific, technical usage in certain branches of mathematics. It does not apply to the everyday use of these words.

For example, take these numbers connected with arrows (i.e. a "graph", another technical term):

There are ways to go around in a closed cycle by following the arrows, e.g. 4 -> 7 -> 9 -> 10 -> 5 -> 4:

Which arrows do we need to remove so that there are no such cycles left?

For example, we could remove 6 -> 2, 10 -> 5, 4 -> 7 to break all cycles. This is a minimal solution because not removing any of these three would leave some cycles intact. Thus the solution can't be made smaller. However, it is not a minimum solution because smaller solutions exist. 5 -> 4, 6 -> 2 would be a minimum solution.

In general, all minimum solutions are also minimal, but the converse is not true.

The problem I described above is called the Feedback Arc Set Problem. You will find several such usages of the words minimal and minimum on its Wikipedia page.

In mathematics, in particular in combinatorics, these words have a very specific contrastive usage as adjectives.

• A minimal solution to a problem can't be made any smaller by "shrinking" it. If we shrink it, it is no longer a solution. There may be other, distinct smaller solutions though.

• A minimum solution to the problem has the smallest possible size among all solutions. No smaller solutions exist.

This is a very specific, technical usage in certain branches of mathematics. It does not apply to the everyday use of these words.

For example, take these numbers connected with arrows (i.e. a "graph", another technical term):

There are ways to go around in a closed cycle by following the arrows, e.g. 4 -> 7 -> 9 -> 10 -> 5 -> 4:

Which arrows do we need to remove so that there are no such cycles left?

For example, we could remove 6 -> 2, 10 -> 5, 4 -> 7 to break all cycles. This is a minimal solution because not removing any of these three would leave some cycles intact. Thus the solution can't be made smaller. However, it is not a minimum solution because smaller solutions exist. 5 -> 4, 6 -> 2 would be a minimum solution.

In general, all minimum solutions are also minimal, but the converse is not true.

The problem I described above is called the Feedback Arc Set Problem. You will find several such usages of the words minimal and minimum on its Wikipedia page.