You have to treat mental health and drug addiction, because other homeless people don't want to live with them. Probably, you'll have to move people out of the city center, if you don't want substantial new taxes. However, if you don't bother to enforce anything you'll still end up with several thousand people living in the creeks under tarps, because they'd rather do that than go outside the city center.
Unless a price of a home is ridiculously cheap (under $100 a month?), and even then, people without any income and/or mentally unwell people will not magically become non-homeless. Next time, read your post two times after writing and check for logical errors.
This sounds like a basic supply and demand question, no?