"Quartering." But these invaders will have less discipline than a formal army, usually.
A ruling from a mayor to force people to take the homeless into their homes would be challenged as unconstitutional under the 3rd amendment.
The Supreme Court ruled that agents of the state are covered by the 3rd amendment.
From Wikipedia Third Amendment to the United States Constitution :
The Third Amendment has been invoked in a few instances as helping establish an implicit right to privacy in the Constitution. Justice William O. Douglas used the amendment along with others in the Bill of Rights as a partial basis for the majority decision in Griswold v. Connecticut (1965), which cited the Third Amendment as implying a belief that an individual's home should be free from agents of the state.
If the mayor demands that the homeless be quartered in private homes, she is declaring them to be wards of the state and a 3rd amendment case could be made.
-PJ