My understanding, from living in Vancouver BC (which also has quite the homeless problem) is that homeless people will migrate from other areas of a country to find the place with the best combination of 1. local weather amenable to living outdoors, and 2. social programs with high ROI (i.e. programs that give you the most benefits for the least bureaucratic folllowing-up.)
No matter how quickly/effectively you deploy solutions for the local homeless problem in a city, of said city is the optimum place for people to live while homeless, you won’t see the end of the problem. (Unless you also have the lobbying power to make changes that solve homeless problems for the entire country.)