Or just build huge amounts of homes. It's not impossible. The US doesn't even build as many dwellings as we used to in the 60s and 70s if I remember correctly.
Enable developers to build skyscrapers of 500 square foot apartments until there are enough of THOSE apartments to house literally 100% of the population. Then people can choose to either pay say $1000/mo to live in one of those small apartments, or whatever it costs to buy a single family home. And if people have the option of renting somewhere cheap, even if it isn't ideal, that'll help balance the market. Right now we have situations where people need to pay $2000+/mo for the smallest of apartments.
You can build huge amount of buildings if you are a societ style central planner and build a ton of cheap prefabs outside of the city. Further urban sprawl doesn't help being of commuting citizens and city ecology.
Or if you are a central planner with the power to evict tenants from parcels. Then it's easy, you just lose credibility as a government.
Or under a system with LVT, the land/housing market clears and unused parcels return to the market, solving the problem by itself with just market design techniques. As you can see, LVT is a necessary component for land/housing market to even function without getting stuck in a feudal monopoly.
Enable developers to build skyscrapers of 500 square foot apartments until there are enough of THOSE apartments to house literally 100% of the population. Then people can choose to either pay say $1000/mo to live in one of those small apartments, or whatever it costs to buy a single family home. And if people have the option of renting somewhere cheap, even if it isn't ideal, that'll help balance the market. Right now we have situations where people need to pay $2000+/mo for the smallest of apartments.