If you look at this problem in terms of bipartite matchings (see
Re: decomposing binary matrices), then your
clean routine is simply removing edges that are never used in any maximum matching (a graph theorist would say: a matching that saturates your set of variables). An efficient way to check this is the following: To check if an edge (u,v) is used in some matching, remove (nodes) u & v from the graph and see if the remaining graph has a maximum matching. I think this is essentially what you are doing, but I'm not sure.
Now, checking a graph for a maximum matching can be done efficiently (see for example http://www.maths.lse.ac.uk/Courses/MA314/matching.pdf). Well, at least much more efficiently than how you have been doing it by trying all possible solutions. Overall, this process for clean would be something like cubic time in the size of your matrix.
I'm away from a perl interpreter at the moment, but can maybe offer something a little more concrete later. Unfortunately, Graph doesn't have any pre-packaged thingy for finding matchings.