Speaking of, the cars are the real problem - the rats proliferate on garbage, of course. So, to cut their food source, the garbage must be hidden from the rats until it is collected: This is axiomatic in my eyes. To hide the garbage, it makes the most sense to just place it inside a modular rat-proof container that is plopped onto a parking space. But it's politically very difficult in NYC to take away parking spaces, especially at a scale that would be required to house all the trash. It's also not possible to go underground like Europe does.
Thus, to truly solve this problem, NYC has to tackle the automobile problem head on and that hasn't been politically popular there for a long time. (Thanks, Robert Moses!) But I'm optimistic for change!