Generalizing the algorithm for domino tiling?

Question

In this earlier question the OP asked the following problem:

Given a rectangular grid where some squares are empty and some are fille

Answer 1

To Keith:

Great work and great explanations! Though, I wrote a program to find maximum tilings, and it uncovered a flaw. Hopefully this can be fixed! [Update: Keith did fix the problem!]

Please check out this link: http://pastebin.com/bABQmfyX (your gadgets analyzed, plus very handy c++ source code)

The problem is that the gadget below can be tiled with 6 dominoes:

y*** ***z
   * *
   ***
   ***
   *x*

-Tom Sirgedas