In what cases is solving Binary Linear Program easy (i.e. **P** complexity)?

The reason I’m asking is to understand if I can reformulate a scheduling problem I’m currently working on in such a way to guarantee finding the global optimum within reasonable time, so any advice in that direction is most welcome.

I was under the impression that when solving a scheduling problem, where a variable value of 1 represents that a particular (timeslot x person) pair is part of the schedule, if the result contains non-integers, that means that there exist multiple valid schedules, and the result is a linear combination of such schedules; to obtain a valid integer solution, one simply needs to re-run the algorithm from the current solution, with an additional constraint for one of the real-valued variables equal to either 0 or 1.

Am I mistaken in this understanding? Is there a **particular** subset of (scheduling) problems where this would be a valid strategy? Any papers / textbook chapter suggestions are most welcome also.