Sort of have the same problem but with horses, I just keep reseeding would help to movethem off certain areas till the seeds take. I rotate my paddocks to regrow some grass in them.

You can either have a nice lawn or dogs. They're pretty much mutually exclusive - especially if it's a small lawn and lots of dog.

Oops. I wasn't finished.
Learn to appreciate the 'moonscape' that the dogs create.
Maybe some nicely-placed garden 'aliens'(in place of gnomes), and faux spacecraft would add a nice touch.
Just be glad you don't have moles, too, - and the huge pits the dogs dig in search of them.