Decision versions of optimization problems: cardinality constraint   (lower bound) as a CNF Horn formula for Maximum Matching
  We provide a formula for the lower bound in the form of $|F| \ge K$, in such a way that the decision version of unweighted non-bipartite matching can be solved in polynomial time. ~The parameter $K$ can vary from instance to instance. We assume that the domains, the set of vertices and the set of edges, are ordered. To our knowledge, no polynomially solvable satisfiability expression has been developed for this problem so far, or for that matter, for any decision problem derived from optimization. Hence for such problems, this opens up a new approach to solving them. 
Prabhu Manyem
cs.LO