Last night the outdoor temperature here was -20 C (-4 F). In conditions like this, it's basic human behaviour to give shelter to those in need. I gladly pay taxes so that the government can solve the problem in the most efficient way for the society. It would be much more expensive to let the ER and health care system handle these people when they get severely frostbitten outside.
This is an actual worst case scenario in our country. If you think we're a little psychotic, it's because if we've experienced poverty in our lives at all, that daily lottery draw has been in the back of our minds since. That sort of insecurity just does not go away.
If you're happy to live without any of those things, then fine -- don't pay taxes.
More importantly you should have read the article until the end. It mentions that the program pays for itself. In other words, the government spends less by providing this housing to the homeless, because the savings due to medical/emergency services are _larger_ than the costs of providing the housing.
From a pure money perspective, it sounds like they cost less to society when they are provided housing, so your concern is somewhat moot if you only look at the monetary side of things (which is, I believe, the angle you were concerned about).