|Welcome to the Monastery|
Re^2: decomposing binary matricesby hv (Parson)
|on Feb 16, 2007 at 17:37 UTC||Need Help??|
Thanks, these are useful terms and concepts for me.
I'm not looking for a perfect match - that would correspond to "a possible assignment". I'm looking for information encapsulating the set of all perfect matches.
One interpretation of that is that if I represent the 5x5 matrix of my OP as a bipartite graph, I'm looking for the edges that are missing from every perfect match, since I can delete them. Deleting those edges would effectively split the graph into connected subgraphs (corresponding to the 2x2 and 3x3 submatrices in the OP).
In short, I'm looking to use the constraint "each variable has a distinct value" to reduce the number of 1s in the matrix.
In practice, I think it would simplify visualisation and make further work more efficient if the resulting discrete subgraphs/submatrices were separated, but I do not think that is in principle necessary - the job is to maximise the information extracted from the constraint. In the example, the constraint tells me that each of the assignments C=2, D=2, B=4 is impossible in any perfect match, so those three 1s in the matrix can be zeroed, or those three edges in the graph can be deleted. Each of the remaining edges can appear in some perfect match, so that's the sum of the information derivable from that constraint at this point.
Update: I should add that in some cases the number of possible values will be greater than the number of variables. In that case, a "perfect match" for the above argument would be a 2n-node cycle that covers each of the n variables: inevitably, some of the values would be left out in any one such match.