NAPLES, Italy — Problems? Yes, conceded Tommaso Sodano, the vice mayor here, Naples has problems. Unpaid debts have reportedly topped $2 billion. Many streets are pocked with potholes. The police department is underfunded, organized crime operates like a shadow state, and illegal dumps are scattered around what is still a grittily beautiful port city.

And then there is what dogs leave behind on the sidewalks.

Naples has no shortage of that, either. Yet to the surprise of some people, including more than a few Neapolitans, the municipal administration is trying to stake out a reputation as a civic innovator by positioning Naples at the cutting edge of dog-waste eradication. By taking DNA samples. Of dogs.

“I know some people find it funny,” Mr. Sodano said, smiling, “that with all the problems the city has, we would focus on dog poop. I know that.”

Well, yes, maybe it is a bit funny. But another thing also appears to be true: For many Neapolitans who must navigate the city’s sidewalks, the initiative is far from unwelcome. In the affluent neighborhood of Vomero, which is serving as a testing ground for the cleanup campaign, many residents are quite pleased, if surprised, that it is happening in Naples.