  We consider $k$-Facility Location games, where $n$ strategic agents report
their locations on the real line, and a mechanism maps them to $k\ge 2$
facilities. Each agent seeks to minimize her distance to the nearest facility.
We are interested in (deterministic or randomized) strategyproof mechanisms
without payments that achieve a reasonable approximation ratio to the optimal
social cost of the agents. To circumvent the inapproximability of $k$-Facility
Location by deterministic strategyproof mechanisms, we restrict our attention
to perturbation stable instances. An instance of $k$-Facility Location on the
line is $\gamma$-perturbation stable (or simply, $\gamma$-stable), for some
$\gamma\ge 1$, if the optimal agent clustering is not affected by moving any
subset of consecutive agent locations closer to each other by a factor at most
$\gamma$. We show that the optimal solution is strategyproof in
$(2+\sqrt{3})$-stable instances whose optimal solution does not include any
singleton clusters, and that allocating the facility to the agent next to the
rightmost one in each optimal cluster (or to the unique agent, for singleton
clusters) is strategyproof and $(n-2)/2$-approximate for $5$-stable instances
(even if their optimal solution includes singleton clusters). On the negative
side, we show that for any $k\ge 3$ and any $\delta > 0$, there is no
deterministic anonymous mechanism that achieves a bounded approximation ratio
and is strategyproof in $(\sqrt{2}-\delta)$-stable instances. We also prove
that allocating the facility to a random agent of each optimal cluster is
strategyproof and $2$-approximate in $5$-stable instances. To the best of our
knowledge, this is the first time that the existence of deterministic (resp.
randomized) strategyproof mechanisms with a bounded (resp. constant)
approximation ratio is shown for a large and natural class of $k$-Facility
Location instances.
