So, a good (not showpiece) solution could have no one place to "put" them. Any solution optimized for the needs of the homeless rather than for their advocates will necessarily be an ever-adapting patchwork based on lots of experiments that are allowed to fail, one that can't ever fully solve the problem but gradually gets better at it, like genuine, evidence-based medicine addressing the needs of sick people.
I don't believe you can constitutionally put someone anywhere they don't want to go unless they threaten others, but you can tell them that they "can't stay here" and let them decide where else to put themselves. Like all of the patches, that will only work for some. Others are unable to put themselves someplace safe, so options have to be created--again different options for different categories. "You can't stay here but you can go here" will be an important patch in the patchwork.
But it still leaves many categories uncovered. Those who are predators can constitutionally be taken off the streets and locked up and need to be. Their most likely prey are other homeless, who need to be kept safe even if you haven't found a housing solution for them. And the mentally ill? There are categories of mental impairments, too. Some aren't too bad off and might be "bribed" in various ways to take their meds.
And I have to repeat, some things will help these people, some things will help those people, and there will always remain unsolved categories. I can no more provide the solution to homelessness than the solution to all illness, but an adapting, experimental, evidence-based approach to solving it in pieces--focused on measured reduction in homelessness rather than on benefits to the service providers--is the most likely to succeed, as is the case with medicine.
A) Where do we put these folks?
B) Can you constitutionally 'put' someone somewhere they may not want to be put?
C) How do you help mentally ill people without resorting to dystopian means?