  Facility location problems often permit facilities to be located at any
position. But what if this is not the case in practice? What if facilities can
only be located at particular locations like a highway exit or close to a bus
stop? We consider here the impact of such constraints on the location of
facilities on the performance of strategy proof mechanisms for locating
facilities.We study four different performance objectives: the total distance
agents must travel to their closest facility, the maximum distance any agent
must travel to their closest facility, and the utilitarian and egalitarian
welfare.We show that constraining facilities to a limited set of locations
makes all four objectives harder to approximate in general.
