Just think about it for a second. Cockroaches are abundant here because there's plenty for them to eat. There's plenty for them to eat because there are a lot of tired, less-than-meticulous people who let dirty dishes pile up in their sinks.
If the city subsidized the cost of dishwashers, and strongly encouraged every apartment in the city to have one installed, those tired, less-than-meticulous people would do their dishes more often, the cockroaches wouldn't have so much to eat, and their numbers would diminish. Not disappear - but I bet we'd cut down the problem.
This is one of those ideas that would make people angry because, you know, why should the city government help people purchase luxury items like dishwashers? But there's a good answer: it would save the city, and all the landlords and homeowners, a lot of money in the long run. Right now the city Health Department has more than 80 full time employees who do nothing but pest control...although apparently they're cutting the number in half for next year. And that's not to mention all the money spent on pest control services.
What makes this idea even better is that it would be popular. Who doesn't want a dishwasher? They make life a lot easier.