In optimization literature, I frequently see solution methods termed "exact" or "approximate". (I use the term "method" here because I suspect exactness, or its lack, is a function of both algorithm and model.) My understanding is that "exact" translates to "produces a provably optimal solution" and "approximate" translates to "may or may not find an optimal solution but will not provide proof of optimality". I'd like to know how others view the terms.
The motivation is a review of recent paper submission. In it, we posed a binary linear programming model and a solution procedure that used branch-and-cut with a variant of Benders decomposition. We called the combination an exact method for solving the underlying problem. We also provided computational examples, all but one of which we solved to optimality. In the one remaining case, I cut off the run at 10 hours with an optimality gap of around 12%. The reviewer pointed to that case as an indication the procedure was not "exact". My initial inclination is to dismiss the comment, since branch-and-cut on a bounded ILP will always reach proven optimality given enough resources (time and memory). On the other hand, from a practical standpoint, if you're not going to get a proven optimal solution within your lifetime, perhaps the method really should not be considered exact.
I know we're not supposed to ask for opinions here, so instead I'll ask which is the proper interpretation of "exact" (or whether I'm completely off base in my interpretation of the term).