The cost may still be a little too far out, but what about requiring breeders to register the dog's DNA sequence in some database. If the dog is found as a stray, a large fine is imposed on the breeder or the most recently registered owner. The fines could go towards programs for neutering and finding homes for strays. If particular breeders are frequently coming up as strays, they could loose their breeder license, etc. Of course to play my own devil's advocate, the risk here is that the owners who have lost interest in keeping a pet kill the animals instead of releasing them.