  We consider a strategic variant of the facility location problem. We would
like to locate a facility on a closed interval. There are n agents located on
that interval, divided into two types: type 1 agents, who wish for the facility
to be as far from them as possible, and type 2 agents, who wish for the
facility to be as close to them as possible. Our goal is to maximize a form of
aggregated social benefit: maxisum- the sum of the agents' utilities, or the
egalitarian objective- the minimal agent utility. The strategic aspect of the
problem is that the agents' locations are not known to us, but rather reported
to us by the agents- an agent might misreport his location in an attempt to
move the facility away from or towards to his true location. We therefore
require the facility-locating mechanism to be strategyproof, namely that
reporting truthfully is a dominant strategy for each agent. As simply
maximizing the social benefit is generally not strategyproof, our goal is to
design strategyproof mechanisms with good approximation ratios.
  For the maxisum objective, in the deterministic setting, we provide a
best-possible 3- approximate strategyproof mechanism; in the randomized
setting, we provide a 23/13- approximate strategyproof mechanism and a lower
bound of \frac{2}{\sqrt{3}}. For the egalitarian objective, we provide a lower
bound of 3/2 in the randomized setting, and show that no bounded approximation
ratio is attainable in the deterministic setting. To obtain our deterministic
lower bounds, we characterize all deterministic strategyproof mechanisms when
all agents are of type 1. Finally, we consider a generalized model that allows
an agent to control more than one location, and provide best-possible 3- and
3/2- approximate strategyproof mechanisms for maxisum, in the deterministic and
randomized settings respectively, when only type 1 agents are present.
