We investigate the complexity of solving stable or perturbation-resilient instances of k-Means and k-Median clustering in fixed dimension Euclidean metrics (or more generally doubling metrics). The notion of stable or 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 solution which remains unchanged 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 in fact finds the OPT solution 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 plausible PCP hypothesis 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.

翻译：我们调查了解决 k- Means 和 k- Median 的稳定性或扰动再恢复情况的复杂性。 我们调查了 k- Means 和 k- Median 的稳定性或扰动性组合在固定维度 Luclide 度( 或更普遍的翻倍度 ) 的参数中的复杂性。 Bilu 和 Linial [2010] 和 Awasthi 等人 [2012] 引入了稳定或扰动性强度实例的概念。 在我们的背景中, k- Means 实例中如果存在一种特殊的地平差( 非一致的), 则该地平地平面会保持不变。 更确切地说, 当距离( 异常的) kMeans 和 kMediarial 算法在实际操作中表现良好时, 该地平面平面的平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平滑。