> They do refuse housing when it is being offered.

This sounds rather incredible... can you back this claim up?

> they do refuse housing we need to do something about it?

Perhaps, but to be honest I'm quite baffled by this. I'm even surprised that San Francisco is offering 0-rent apartments to the homeless, when generally from what I know the rental market there is rather out of control.

> by at least deporting illegal aliens?

the problem you decried is people who are homeless and harass others on the streets. So, why use someone's legal immigration status as the criterion for action? I'd imagine most homeless are actually US citizens, and that illegal immigrants tend to find someplace to work and some decent sleeping arrangement.

If you build enough shelter space to house the homeless population, by law your choices are to sleep in the shelter or go to jail if you refuse. Until enough shelter space is built, you can continue to sleep on the sidewalk. See the Boise decision which dictates this policy all over the Western United States.