We study the classic facility location setting, where we are given n
clients and m possible facility locations in some arbitrary metric space, and
want to choose a location to build a facility. The exact same setting also
arises in spatial social choice, where voters are the clients and the goal is
to choose a candidate or outcome, with the distance from a voter to an outcome
representing the cost of this outcome for the voter (e.g., based on their
ideological differences). Unlike most previous work, we do not focus on a
single objective to optimize (e.g., the total distance from clients to the
facility, or the maximum distance, etc.), but instead attempt to optimize
several different objectives simultaneously. More specifically, we consider the
l-centrum family of objectives, which includes the total distance, max
distance, and many others. We present tight bounds on how well any pair of such
objectives (e.g., max and sum) can be simultaneously approximated compared to
their optimum outcomes. In particular, we show that for any such pair of
objectives, it is always possible to choose an outcome which simultaneously
approximates both objectives within a factor of 1+2β, and give a
precise characterization of how this factor improves as the two objectives
being optimized become more similar. For q>2 different centrum objectives, we
show that it is always possible to approximate all q of these objectives
within a small constant, and that this constant approaches 3 as qββ. Our results show that when optimizing only a few simultaneous
objectives, it is always possible to form an outcome which is a significantly
better than 3 approximation for all of these objectives.Comment: To be published in the Proceedings of 37th Conference on Artificial
Intelligence (AAAI 2023