Often, we can model combinatorial optimization problems with an Integer Program. Then there is an associated Linear Relaxation which drops the integrality constraints on the variables.
Let's say we are dealing with a maximization problem. Then the Integrality Gap of a Linear Relaxation is the maximum ratio between the optimum LP solution and the optimum IP solution.
Suppose that I have a Linear Relaxation for a problem and I am able to prove an upper bound of $\alpha$ on the Integrality Gap for all problem instances. Furthermore, I am able to solve the Linear Relaxation in polynomial time.
Does this imply that there is an $\alpha$-approximation for my original combinatorial problem?
Remark: Often the proof of the upper bound on the integrality gap involves giving an explicit rounding technique which can then be converted into an approximation algorithm. But my question is, is there a general procedure to go from a bounded integrality gap to an approximate solution to the integer program?