Yes, the city has a responsibility to work on the local level to find safe places for those without housing. But, really, this is such a massive problem that it really needs to be a joint activity, state, city, county, and federal government. At a minimum the police need to be told that homelessness is not a crime, and if the city cannot point to a place for a poor person to sleep, that poor person has a right to survive.
Miami Florida worked to  solve the problem with a meals tax and other funding sources, but only after a successful lawsuit that forced the county to allow all homeless people live as best they could without police interference. 
https://aclufl.org/resources/pottinger-agreement-faq/
No bed, no arrest. 
No public shelter, no arrest. 
I thank the RG for published this series on homelessness. It may educate many to the weird absence of affordable housing, even at the Pubic Housing level. State legislators seem absent on this. 
Hugh Massengill, Eugene Oregon