Binary integer programming problem without exponential memory
There is in general not likely to be any algorithm for this problem that will take polynomial time (and polynomial space); the problem is NP-hard, and thus not believed to allow an efficient algorithm.  If you just want polynomial space, then exhaustive search works.  You're not likely to find any algorithm that uses polynomial space and is guaranteed to be faster than exhaustive search; that would contradict the strong exponential-time hypothesis.  All existing methods are heuristics that might work well on some instances but not others.