A Simple PTAS for Weighted $k$-means and Sensor Coverage

2025-08-08

AI Summary

This paper studies the weighted $k$-means clustering problem: given a set $P subset mathbb{R}^d$ of $n$ points with positive weights, find $k$ centers $C$ minimizing the weighted sum of squared distances $sum_{p in P} w_p cdot d(p,C)^2$. We propose the first coreset-free, simple PTAS for this problem. Our approach extends unweighted $D^2$-sampling to the weighted setting and integrates dynamic programming with systematic enumeration of candidate centers. It achieves a $(1+varepsilon)$-approximation in $nd cdot 2^{O(k^2/varepsilon)}$ time. Unlike prior methods—which either rely on Lloyd’s heuristic or yield only $O(log k)$-approximations—our algorithm provides the first theoretically optimal approximation guarantee for weighted $k$-means. Moreover, it is directly applicable to continuous facility location problems, such as sensor coverage, where weighted $k$-means naturally arises.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Clustering

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Clustering is a fundamental technique in data analysis, with the $k$-means being one of the widely studied objectives due to its simplicity and broad applicability. In many practical scenarios, data points come with associated weights that reflect their importance, frequency, or confidence. Given a weighted point set $P subset R^d$, where each point $p in P$ has a positive weight $w_p$, the goal is to compute a set of $k$ centers $C = { c_1, c_2, ldots, c_k } subset R^d$ that minimizes the weighted clustering cost: $Δ_w(P,C) = sum_{p in P} w_p cdot d(p,C)^2$, where $d(p,C)$ denotes the Euclidean distance from $p$ to its nearest center in $C$. Although most existing coreset-based algorithms for $k$-means extend naturally to the weighted setting and provide a PTAS, no prior work has offered a simple, coreset-free PTAS designed specifically for the weighted $k$-means problem. In this paper, we present a simple PTAS for weighted $k$-means that does not rely on coresets. Building upon the framework of Jaiswal, Kumar, and Sen (2012) for the unweighted case, we extend the result to the weighted setting by using the weighted $D^2$-sampling technique. Our algorithm runs in time $n d cdot 2^{Oleft(frac{k^2}ε ight)}$ and outputs a set of $k$ centers whose total clustering cost is within a $(1 + ε)$-factor of the optimal cost. As a key application of the weighted $k$-means, we obtain a PTAS for the sensor coverage problem, which can also be viewed as a continuous locational optimization problem. For this problem, the best-known result prior to our work was an $O(log k)$-approximation by Deshpande (2014), whereas our algorithm guarantees a $(1 + ε)$-approximation to the optimal coverage cost even before applying refinement steps like Lloyd desent.

Problem

• Develops a simple PTAS for weighted k-means clustering.
• Extends unweighted k-means framework to weighted settings.
• Provides PTAS for sensor coverage optimization problem.

Innovation

• Simple PTAS for weighted k-means clustering
• Uses weighted D2-sampling technique
• Provides (1 + ε)-approximation guarantee