We will investigate the special case of the restricted assignment with 2 values, but the method potentially works for other problems as well.
The goal is the following: instead of considering the real integrality gap (currently known to be at most 5/3), we implement the idea of core instances. Given an error tolerance epsilon, we discretize the set of all potential inputs with the epsilon-grid.
On this finite set, we calculate the gap g (for small dimensions, say n = 10, m = 4), and by this we are able to derive an upper bound of g + (n x epsilon) for the "real gap" in this small dimensions.
For a small enough epsilon, we hope to gain further insight on the "real gap" (so far known to be between 3/2 and 5/3) by examining the experimental bound. If gap + (n x epsilon) is smaller, we could state a conjecture that the "real gap" is that specific value (provided that we are able to determine it), and try coming up with better approximation algorithms. On the other hand, if the "empirical gap" seems close to 5/3, the conjecture could be that the real gap is 5/3 as well, and researchers could channel efforts to find instances with this specific gap.
Emprically approximating the gap could point us in the direction of either design better approximating algorithms, or constructing instances with higher integrality gaps than the current champion (3/2).