Clustering in Varying Metrics

2025-10-09

AI Summary

This paper studies the multi-metric agglomerative center clustering problem: given a set of $n$ points represented under $T$ distinct metrics, select $k$ centers to minimize the $Psi$-norm (e.g., $ell_1$, $ell_infty$) of their $k$-center/median/means costs across all metrics. We first establish that no finite approximation ratio exists for $T geq 3$. For $T = 2$, we design constant-factor approximation algorithms. We introduce a joint parameterized framework yielding a 3-approximation when both $k$ and $T$ are bounded; achieve a $(1+varepsilon)$-approximation for instances with bounded $varepsilon$-scattering dimension or treewidth; and prove, under the Exponential Time Hypothesis (ETH), that no nontrivial approximation is possible when parameterized solely by $T$. Our results provide a unified characterization of the computational boundaries and tractable structures in multi-metric clustering.

Technology Category

• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Clustering

Application Category

• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce the aggregated clustering problem, where one is given $T$ instances of a center-based clustering task over the same $n$ points, but under different metrics. The goal is to open $k$ centers to minimize an aggregate of the clustering costs -- e.g., the average or maximum -- where the cost is measured via $k$-center/median/means objectives. More generally, we minimize a norm $Ψ$ over the $T$ cost values. We show that for $T geq 3$, the problem is inapproximable to any finite factor in polynomial time. For $T = 2$, we give constant-factor approximations. We also show W[2]-hardness when parameterized by $k$, but obtain $f(k,T)mathrm{poly}(n)$-time 3-approximations when parameterized by both $k$ and $T$. When the metrics have structure, we obtain efficient parameterized approximation schemes (EPAS). If all $T$ metrics have bounded $varepsilon$-scatter dimension, we achieve a $(1+varepsilon)$-approximation in $f(k,T,varepsilon)mathrm{poly}(n)$ time. If the metrics are induced by edge weights on a common graph $G$ of bounded treewidth $mathsf{tw}$, and $Ψ$ is the sum function, we get an EPAS in $f(T,varepsilon,mathsf{tw})mathrm{poly}(n,k)$ time. Conversely, unless (randomized) ETH is false, any finite factor approximation is impossible if parametrized by only $T$, even when the treewidth is $mathsf{tw} = Ω(mathrm{poly}log n)$.

Problem

• Aggregated clustering under multiple metrics for same points
• Minimizing norm of costs across T different clustering instances
• Developing approximations for varying metrics with structural constraints

Innovation

• Aggregated clustering across multiple metric instances
• Parameterized approximations for bounded scatter dimensions
• Efficient schemes for treewidth-bounded graph metrics