Yeah, joking aside I think pet registry has to be the other side of this + enforcement with DNA tests. If you have puppies, and they end up as strays, there should be sanctions.
Depends how affordable DNA tests are. For a developing country, sterilization and vaccination might be cheaper than tracking every stray dog to their or their parent's owner. Policy can also make this infeasible if, for example, dog registration is too expensive in the first place for owners.
It might be surprisingly cheap. Prague was doing DNA matching on dog poop in the streets to fine owners for not cleaning up.
I think there were some problems from people breaking the poop bins and scattering the contents everywhere to reduce trust in the system ("it's not my fault, I cleaned up, someone took it out of the bin") but that wouldn't be a problem for strays right?
The Czech Republic is not a developing country though, but the wealthiest of the former Eastern Block countries. The cost might be easily covered by the fines, but I'm not so sure whether enough money can be extracted from perpetrators in less affluent countries.