[yitosda]

Depends on how you define "the problem". Homelessness is a symptom of several other things widely considered to be problems.

If you limit the problem to "some people who want a house don't have enough money to afford living space qualifying as a house", then broadly speaking market solutions are possible because we can assume these people participate in the market via their "demand" and their ability to "supply" something in return. Obviously there's an active debate on many market issues here, whether around too much market regulation (eg NIMBY), or too little market regulation (eg minimum wage).

However, my understanding is that there is mounting evidence that the homeless population is composed of people who have either no demand for a house ("free spirits") or no ability to supply anything to the market (mentally disabled). Obviously in these cases the market is not very useful, and only functions insomuch as those outside of the homeless population include it in their "demand function".