How Finland avoided a homelessness crisis
Apparently the key is to give them free housing and tailored support services and to involve the community. Eliminating homelessness has strong support in Finland, and per a BBC article, "the average minimum winter temperature in Helsinki is -7C".
Including information from other sources, it looks like about 80% stay off the street, and the cost was a little under $100k per unit over a decade including support services. Supposedly the program pays for itself in savings in services up to $11k/yr or "society's funds" of $17k/yr (Y-Foundation). Y-Foundation believes the model can be replicated.