in reply to decomposing binary matrices

As I suggested in the CB, if your ultimate goal is only to find an assignment of values to variables that meets all the constraints, then you don't have to actually decompose the matrix. All you need to do is find a perfect bipartite matching.

A bipartite graph is a graph where the vertices are partitioned into 2 sets, and every edge crosses between these sets (i.e, does not stay within a set). In this case, one set denotes your variables and the other set denotes your values. Draw an edge between a variable and a value if it is legal to assign that value to that variable.

A matching is a subset of edges from a graph, where no two edges share an endpoint. Since all edges in a bipartite graph run between these 2 sets, you can think of a matching as a (partial) mapping from 1 set to the other (each vertex in set A is paired with at most 1 vertex in set B). In this case, it represents a mapping from variables to values. Since the matching is a subset of the available edges, the mapping doesn't include any assignments that you have deemed "illegal".

What you want is a perfect matching, one that covers all vertices (so it's a 1-to-1 mapping between variables and values). Usually this problem is phrased as the "marriage problem" -- The two sets of vertices are men & women, and there is an edge between a man & a woman if they could be happy married. You want to know if there is a way to pair them up so that each couple is happy (and if there is such a way to pair them up, output it).

An example that I found with pictures is here: bipartite matching. Googling for bipartite matching algorithm turns up plenty of hits. I know that there are efficient algorithms for doing it, but can't remember all of the details.