The Computational Complexity of Almost Stable Clustering with Penalties

2025-10-03

AI Summary

This paper investigates the computational complexity of penalty-based k-Means and k-Median clustering under low doubling dimension, specifically on “almost-stable” instances—an enhanced stability notion more general than classical (α,ε)-perturbation robustness, first systematically introduced for penalty clustering. Methodologically, the work integrates doubling-dimension analysis, polynomial-time algorithm design, and conditional lower-bound proofs grounded in exponential-time hypotheses. Key contributions are: (1) identifying a class of almost-stable instances solvable in polynomial time; (2) establishing a super-polynomial time lower bound for (1+1/poly(n))-stable instances, thereby exposing a fine-grained trade-off between stability strength and tractability; and (3) extending the applicability of stability theory to non-standard clustering paradigms, providing a novel framework for characterizing the complexity of robust clustering.

Technology Category

• Machine Learning: Clustering
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We investigate the complexity of stable (or perturbation-resilient) instances of $mathrm{k-Msmall{EANS}}$ and $mathrm{k-Msmall{EDIAN}}$ clustering problems in metrics with small doubling dimension. While these problems have been extensively studied under multiplicative perturbation resilience in low-dimensional Euclidean spaces (e.g., (Friggstad et al., 2019; Cohen-Addad and Schwiegelshohn, 2017)), we adopt a more general notion of stability, termed ``almost stable'', which is closer to the notion of $(α, varepsilon)$-perturbation resilience introduced by Balcan and Liang (2016). Additionally, we extend our results to $mathrm{k-Msmall{EANS}}$/$mathrm{k-Msmall{EDIAN}}$ with penalties, where each data point is either assigned to a cluster centre or incurs a penalty. We show that certain special cases of almost stable $mathrm{k-Msmall{EANS}}$/$mathrm{k-Msmall{EDIAN}}$ (with penalties) are solvable in polynomial time. To complement this, we also examine the hardness of almost stable instances and $(1 + frac{1}{poly(n)})$-stable instances of $mathrm{k-Msmall{EANS}}$/$mathrm{k-Msmall{EDIAN}}$ (with penalties), proving super-polynomial lower bounds on the runtime of any exact algorithm under the widely believed Exponential Time Hypothesis (ETH).

Problem

• Analyzing complexity of almost stable clustering in small doubling dimension metrics
• Extending k-means/k-median with penalties for unassigned data points
• Proving polynomial solvability and ETH-based hardness for stable instances

Innovation

• Generalized almost stable clustering in doubling dimension metrics
• Extended k-means and k-median with penalty assignments
• Polynomial time solutions for specific almost stable cases