Is the correct phrase "free run" or "free roam?"
Solution 1:
"Free run" seems to be the hands down winner, with "free roam" barely registering as a phrase.
("Free range" is another strong contender.)
Solution 2:
The expression I know is "free run of" meaning the dog (in your case) can make full use of the house. But I've heard it without the adjective "free", too.
Solution 3:
"free roam" is the most common in my experience (southern UK), but usually without qualification (i.e. "the dog had free roam" rather than "the dog had free roam of the house)