  We investigate the complexity of solving stable or perturbation-resilient
instances of $k$-Means and $k$-Median clustering in fixed dimension Euclidean
metrics (more generally doubling metrics). The notion of stable (perturbation
resilient) instances was introduced by Bilu and Linial [2010] and Awasthi et
al. [2012]. In our context we say a $k$-Means instance is $\alpha$-stable if
there is a unique OPT which remains optimum if distances are (non-uniformly)
stretched by a factor of at most $\alpha$. Stable clustering instances have
been studied to explain why heuristics such as Lloyd's algorithm perform well
in practice. In this work we show that for any fixed $\epsilon>0$,
$(1+\epsilon)$-stable instances of $k$-Means in doubling metrics can be solved
in polynomial time. More precisely we show a natural multiswap local search
algorithm finds OPT for $(1+\epsilon)$-stable instances of $k$-Means and
$k$-Median in a polynomial number of iterations. We complement this result by
showing that under a new PCP theorem, this is essentially tight: that when the
dimension d is part of the input, there is a fixed $\epsilon_0>0$ s.t. there is
not even a PTAS for $(1+\epsilon_0)$-stable $k$-Means in $R^d$ unless NP=RP. To
do this, we consider a robust property of CSPs; call an instance stable if
there is a unique optimum solution $x^*$ and for any other solution $x'$, the
number of unsatisfied clauses is proportional to the Hamming distance between
$x^*$ and $x'$. Dinur et al. have already shown stable QSAT is hard to
approximate for some constant Q, our hypothesis is simply that stable QSAT with
bounded variable occurrence is also hard. Given this hypothesis we consider
"stability-preserving" reductions to prove our hardness for stable k-Means.
Such reductions seem to be more fragile than standard L-reductions and may be
of further use to demonstrate other stable optimization problems are hard.
