Suppose we have an LP with rational coefficients.
To my knowledge, this implies that the optimal solution to that LP is also rational. In other words, every variable may be written as:
$$x_{i}^{\star} = M_{i}/N_{i}$$
Moreover, I can scale numerator/denumerator such that all variables share the same denumerator:
$$x_{i}^{\star} = \frac{M_{i}^{\prime}} {N_{common}}$$
Is it possible to find a common denumerator $N_{common}$ before solving the LP? This would convert the optimization problem to a problem which must have an integer solution (optimizing over variables $M_{i}$.
Another interpretation of my question: Is it possible to solve any LP by having each variable take values which are multiples of $1/N_{common}$, with the latter having the notion of the required "precision" to avoid losses?