I know that the exact cover problem has a pseudopolynomial algorithm when the cover size is a given constant (as here: Is set cover still NP-complete if you have a given k?).
However, I am interested in approaches to solving a variant of Exact cover (or rather, twice-exact cover, that I asked about here: Solving a variant of the Exact cover problem) that has a solution of a known size that is not a constant, but rather of a known size that is $O(n)$.
The problem arises from reconstruction of the faces of (perhaps non-convex) Polyhedra given their vertices and edges (as detailed in the comments to the linked question), and the known size arises from Euler's formula for Polyhedra $V-E+F=2$ (and therefore I am looking for an twice-exact set cover of the set of edges, with size $F=E-V+2$).
what kind of approaches exist for solving such a problem? (I have thought of some Linear programming approach, but I have had difficulties formulating the conditions, and I do not know how fast would it run).
Thank you in advance.
