When we consider an approximation algorithm for a minimization problem, the integrality gap of an IP formulation for this problem gives a lower bound of an approximation ratio for certain class of algorithms (such as rounding or primal-dual algorithm). In fact, there are many problems whose best approximation ratio matches the integrality gap.
Some algorithm may have a better approximation ratio than the integrality gap for some problem, but I don't know if such an example exists or not. If the answer is yes, could you give some examples?
I know that some problems admit multiple mathematical formulations. In such cases, consider the mathematical formulation with the smallest integrality gap, as long as it can be solved in polynomial time (perhaps some formulations may use separation oracles).
This question is related to [the question: The importance of Integrality Gap].