Combining inequalities to not have coefficients

Question

We have in input an inequality cons and a set of inequalities C and we want to find a way to sum them and simplify in a way that no variable has a coefficient and that the number of constraints that we took is maximal.

The input is always a constraint without any coefficient and it is always a <= k for a given k.

For example we can have in input:

cons = x1 + x2 + x3 <= 1
C:
   c1 = x1 + x2 + x4 <= 1
   c2 = x1 + x5 <= 1
   c3 = x3 + x4 <= 1

• If we sum: cons + c1, we obtain 2x1 + 2x2 + x3 + x4 <= 2 and we do not respect the condition that no variable should have a coefficient.

• If we sum cons + c2, we obtain 2x1 + x2 + x3 + x5 <= 2, so again it is not working.

• If we sum cons + c3, we obtain x1 + x2 + 2x3 + x4 <= 2, so again it is not working.

• However, if we sum cons + c1 + c3, we obtain 2x1 + 2x2 + 2x3 + 2x4 <= 3, thus it is x1 + x2 + x3 + x4 <= floor(3/2), and this time it works.

Does it exist an optimal way to solve this problem except going through all the possibilities?

Does this problem has a name in the literature?