Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means

TL;DR

This paper investigates perturbation-resilient ($\alpha$-stable) instances of $k$-means and $k$-median in Euclidean and doubling metrics. It shows that for any fixed $\epsilon>0$, $(1+\epsilon)$-stable instances in fixed dimension admit polynomial-time solutions via a best-improvement $\rho$-swap local search, and that this tractability is essentially tight under a stability-preserving PCP framework, yielding hardness in high dimensions. The hardness proof builds a stability-preserving reduction chain from S-QSAT-B to stable variants of 3SAT, 3D matching, and a covering-by-triples problem, culminating in a Euclidean $k$-means instance with a provably stable optimum. A key methodological contribution is the notion of stability-preserving reductions, which are more delicate than standard L-reductions but enable transferring stability properties across problem domains. Collectively, the results deepen understanding of when clustering can be solved exactly or approximated efficiently under perturbation resilience and offer a blueprint for proving hardness for other stable optimization problems.

Abstract

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 $α$-stable if there is a unique OPT which remains optimum if distances are (non-uniformly) stretched by a factor of at most $α$. 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 $ε>0$, $(1+ε)$-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 $(1+ε)$-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 $ε_0>0$ s.t. there is not even a PTAS for $(1+ε_0)$-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.

Figures (2)

• Figure 1: The variable gadget for a variable $x_i$ with $\beta_i = 4$.
• Figure 2: Example of part of a clause gadget for a clause $C = \overline{x_i} \vee x_j \vee \overline{x_k}$ where $b_i, b_j$ and $b_k$ denote which occurrence of the corresponding variable appears in $C$. Here, $\alpha$ is the assignment $x_i = \texttt{True}, x_j = \texttt{False}, x_k = \texttt{False}$ which satisfies $C$. In particular, the tips of the gears that appear in this figure correspond to $\alpha$, not necessarily to the sign of the original variable in $C$. The rounded rectangles indicate the four triples in this gadget associated with $\alpha$.

Theorems & Definitions (49)

• Definition 1: $\alpha$-stability
• Theorem 1
• Definition 2: Unambiguous QSAT
• Theorem 2
• Theorem 3
• Definition 3: Stable SAT Instances
• Definition 4: S-QSAT-B
• Theorem 4
• Definition 5
• Theorem 5