What kind of policies you believe would affect rents in the short to medium term? Progressive cities have tried all kinds of things, with little in terms of results.

Short and medium? That's probably a tough ask besides subsidiaries. But also allow fast redevelopment of office space. In Germany there was and is the "Munich" effect. Zone too many office districts so office space is cheap and companies move there. So also more mixed use zoning for white color work places for short commutes

So what? government just orders lower rents for current renters? That might work short term, but in the longer one it will simply destroy the local property market. No one will be willing to build housing anymore, as it won't be profitable. The city will slowly gentrify and die.

As I posted below and also public transport as mentioned in other comments

Change zoning rules. Incentives for co-living. Reduce flat sizes. Rebuilt office spaces. Or look to Vienna: a big stock of goverment-owned flats with low prices which makes it hard to charge more than those.

In Berlin they did this with the "Mietendeckel" and the result was unsurprising (supply shortages, black markets).

They at least tried, but also it was abbolished bc it was unconstitutional. So the current market is not bc of it.

The current market still suffers from other forms of government regulation. The Mietendeckel still exists in some form via the Mietenspiegel, and basically the entire city under what Americans understand as rent-control.

How does it suffer to make it illegal to charge 10% more or less than the average for comparable housing? It seems to make it more stable and less volatile.

Go look up the Mietenspiegel price for your flat in your neighborhood, then pop open Immoscout and try and find one for that price.

Price controls = supply shortages. Basic principal of market economics.

But how does the market suffer from that? It implies that the Mietenspiegel is a bad policy. It only protects from price gauging or price dumping. If you are living in Berlin I would highly recommend looking up if you're paying too much.

Lines of 50 people to see an apartment sounds like a huge waste of man hours, which cost money to economy. This waste also grows quadratically: more people searching for longer.

Which government policies exactly would you suggest that don't involve building more houses or convincing people to move elsewhere?

Remember, this is Switzerland, so you need to convince the population that this is a good idea otherwise they will block it with a popular vote. So confiscation of property or forced labour or increased taxes are all out of the question.

I agree I should have written a more constructive comment with more details. But I thought it was clear how policy can help in the short/medium/long-term. Please read the answers I wrote to the commenters below/above