Update 6/1 - I personally tested burying poop and pouring vinegar over the dig holes. I also sprinkled used coffee grounds are areas that were prone to digging. The digging has stopped for us. On to the next thing...
Why dogs dig and how to save your yard from ruin...
If you could ask your dog why they love to dig so much, you might get one of the following answers:
So it is easy enough to understand why they love to dig. A tougher question is what do you do about it once it has begun? Digging is very self-rewarding. How do you get your dog to stop doing something they clearly enjoy so much?
Have a designated place to dig. If you can give them one spot that is just for digging, It is easy enough to teach them that this is the only spot to enjoy the digging. A kiddie pool filled with dirt is great. You can even bury some toys in the dirt to make it even more exciting.
Redirect and reward with activities that are even more fun. Using positive interrupters are a great way to train dogs. https://www.youtube.com/watch?v=TBvPaqMZyo8t
Exercise is a great way to occupy your dog's body in a positive way. A tired dog will be less inclined to dig up the yard.
Mental stimulation is a must for most dogs. The smarter the dog, the more you will need to stimulate them mentally. Obedience training is great for mental stimulation. Training sessions will tire a dog out as much as a walk - just in a different way.
Monitor behavior. Catch the digging early. A firm "no dig" and re-direction helps. Reward good behavior. Bury chicken wire in the holes and cover with rocks to make the digging areas less desirable temporarily. Make sure to roll sharp edges away from digging surface. If the pet is intent on escaping, a concrete border poured along the perimeter and at least 12" deep may be necessary to keep some breeds from digging their way to freedom. As with all habits, the longer it is allowed to persist, the harder it will be to correct.
In addition to the previous tips, the one that is often the quickest and easiest is to bury the dog's poop in the holes. Fill with some cheap vinegar. I like to stock up at the dollar stores. Dogs don't like the smell of vinegar and most do not want to dig in poop. Once the holes are filled, cover them with dirt. You won't smell the poop once it is covered. But, your dog will definitely smell it. You will have to be consistent. After a couple weeks of this, most dogs will give up and go onto something that smells better to occupy their time. As with all things concerning dogs, patience and persistence is the most important quality for a dog owner to possess.