The European house mouse is very distantly related to its cousin the Algerian mouse. Many of the hybrids that form are not viable. However, there are some exceptions: sometimes, when the European house mouse and the Algerian mouse mate, they produce offspring with the advantageous gene that gives them a distinct advantage: resistance to the pesticide warfarin. Surely, this must be an issue of concern. And I don't think bringing cats into the picture will eliminate mice from households.