It's happening all over Germany. They are getting the apartments paid by the state, so high rents are being paid for them. 1-2 million is a huge number of people if you're lacking living space. Where to put them?
- highly skilled workers (IT, Finance, Sales, Marketing) that earn much more than average german folk, and compete for housing with upper class berliners
- unskilled workers (constructions) that live usually in temp housing and don't attempt to rent a flat for themselves.
As for the refugees, most of them still live in camps distributed across Germany and if they get a flat subsidised by the state it will be in some low demand neighbourhood.
Seriously though in the UK we have had immigration over the last 20 years and that must have led to higher demand and prices. Best estimates are about 20% in 20 years in England.
https://fullfact.org/immigration/have-house-prices-risen-bec...
BUT
English house prices have risen by 320% on average over this period.
Other factors are - getting married later, more single people, more divorced couples, and financialisation of housing.
Probably a similar story in Berlin.
However, since the cheap money from the banks, more people are orienting to buy than rent, which is quite different from the past.