Let S be a set of integers. Now which of the following sentences are correct:
The set S contains ...
Set S contains ...
None of those are grammatically incorrect, but I prefer the last one. You have already defined S as a set, so the word set is superfluous. That said, mentioning that S is a set could be warrented if it would serve as a useful reminder. For example, suppose you have a lot of one-letter variables you are discussing in one section of your paper.
Let S be the set of odd prime numbers. Let M be an 8x8 matrix such that each value is an even integer. Let H be a closed Hamiltonian graph. Let F = 96,485 J (Faraday's constant). The set S contains...
In that case, a reminder that S is a set may be justified, because single upper-case letters are being used to represent sets, graphs, constants, and matrices. A small reiteration helps the reader remember what we are talking about. However, in a paper where each capital letter denotes a set:
Let S be the set of odd prime numbers. Let T be the set of squared integers. Let R be the set of integers found in the Fibonacci sequence. S contains...
I don't see any need to specify that S is a set at the start of the sentence. However, that's a stylistic choice, not a grammatical requirement.
As for your sentence:
By the assumption, facility i is fully paid before time t. For any city k, the edge j must be tight before time t.
Here's my advice: Don't overthink it. I don't think either of these would be more grammatically correct or incorrect:
By the assumption, the facility i is fully paid before time t. For any city k, the edge j must be tight before time t.
By the assumption, facility i is fully paid before time t. For any city k, edge j must be tight before time t.
Sometimes, the inclusion of an article is absolutely necessary:
I bought car yesterday. I bought a car yesterday.
Other times, the inclusion of an article is purely optional, and mathematics provides plenty of examples of that:
Let S be the set of all prime numbers; let R be the set of all even integers greater than zero. Let T be the intersection of R and S.
The set T has exactly one member.
T has exactly one member.
Set T has exactly one member.
All three of those conclusions are stated in a way that is grammatically correct. The definite article (the) is acceptable, because there is only one set T. However, the definite article isn't necessary, because the set already has a unique name (T).
I've already explained which one of the three I prefer.