In what cases is solving Binary Linear Program easy (i.e. **P** complexity)? I'm looking at scheduling problems in particular
It is not true that fractional solutions can necessarily be turned into integral solutions.  For example, suppose that 1 requires either slots A and B, or slots D and E; 2 requires either slots B and C, or slots E and F; and 3 requires either slots A and C, or slots D and F.  It's easy to check there is no integral solution at all (A,B,C together can accommodate at most one, and D,E,F together can accommodate at most one).  However there is a fractional solution of giving everyone 1/2 of every slot that might possibly accommodate them.
There are conditions such as total unimodularity that guarantee that there are optimal integral solutions.  In that case, the problem is in P, because all that is needed is to solve the LP relaxation which can be done in polynomial time.