  The study of approximate mechanism design for facility location problems has
been in the center of research at the intersection of artificial intelligence
and economics for the last decades, largely due to its practical importance in
various domains, such as social planning and clustering. At a high level, the
goal is to design mechanisms to select a set of locations on which to build a
set of facilities, aiming to optimize some social objective and ensure
desirable properties based on the preferences of strategic agents, who might
have incentives to misreport their private information such as their locations.
This paper presents a comprehensive survey of the significant progress that has
been made since the introduction of the problem, highlighting the different
variants and methodologies, as well as the most interesting directions for
future research.
