Homelessness has recently risen rapidly in cities where the cost of rent has also recently risen rapidly.

So is it that other cities are suddenly shipping over way more people, or is it a predominantly local problem caused by rapidly rising rent that can no longer be afforded?

We have cities where existing residents can no longer afford housing that are actively seeking to attract more business to move in, and with that create more competition for housing and higher population. You can't fix the results of housing cost inflation if you keep the upward pressure on the cost - you just will be spending more of your money chasing the upwardly-moving property prices your other policies are causing!