[Paper Review] Faster Balanced Clusterings in High Dimension

Hu Ding|arXiv (Cornell University)|Sep 4, 2018

Facility Location and Emergency Management47 references3 citations

TL;DR

This paper presents a novel geometric framework for faster balanced clustering in high dimensions, using spatial partitioning to simplify and accelerate solutions for balanced $k$-center, $k$-median, and $k$-means problems. It achieves $4$-approximation for $k$-center and constant or $(1+ ho)$-approximation for $k$-median and $k$-means with linear or nearly linear time when $k$ is constant, significantly improving prior methods.

ABSTRACT

The problem of constrained clustering has attracted significant attention in the past decades. In this paper, we study the balanced $k$-center, $k$-median, and $k$-means clustering problems where the size of each cluster is constrained by the given lower and upper bounds. The problems are motivated by the applications in processing large-scale data in high dimension. Existing methods often need to compute complicated matchings (or min cost flows) to satisfy the balance constraint, and thus suffer from high complexities especially in high dimension. We develop an effective framework for the three balanced clustering problems to address this issue, and our method is based on a novel spatial partition idea in geometry. For the balanced $k$-center clustering, we provide a $4$-approximation algorithm that improves the existing approximation factors; for the balanced $k$-median and $k$-means clusterings, our algorithms yield constant and $(1+ε)$-approximation factors with any $ε>0$. More importantly, our algorithms achieve linear or nearly linear running times when $k$ is a constant, and significantly improve the existing ones. Our results can be easily extended to metric balanced clusterings and the running times are sub-linear in terms of the complexity of $n$-point metric.

Motivation & Objective

• To address the computational inefficiency of existing balanced clustering algorithms that rely on complex matchings or min-cost flows in high-dimensional spaces.
• To develop a unified, geometry-based framework that simplifies and accelerates solving three key balanced clustering problems: $k$-center, $k$-median, and $k$-means.
• To achieve provably good approximation ratios while reducing time complexity, especially for constant $k$.
• To extend the framework to metric spaces and ensure sub-linear or near-linear runtime in terms of $n$.

Proposed method

• Introduces a novel spatial partitioning technique in high-dimensional geometry to decouple cluster assignment from balance enforcement.
• Uses Chen's bi-criteria $k$-median/$k$-means algorithm to generate $O(k)$ candidate centers, reducing the search space.
• Employs linear programming (LP) relaxation with a rounding procedure (Theorem 2.2) to convert fractional solutions into integral cluster assignments while respecting size bounds.
• Leverages spread ratio $\Delta$ and $\epsilon$-dependent sampling to bound distance ratios and ensure $(1+\epsilon)$-approximation for $k$-median and $k$-means.
• Adapts techniques from [14] and [26] to generate candidate $k$-tuples of centers with high probability of yielding $(1+\epsilon)$-approximate solutions.
• Combines geometric insights with LP-based rounding to maintain approximation guarantees while minimizing computational overhead.

Experimental results

Research questions

• RQ1 Can a geometric spatial partitioning approach replace complex matching or min-cost flow computations in balanced clustering?
• RQ2 Can the framework achieve constant or $(1+\epsilon)$-approximation for $k$-median and $k$-means with near-linear time complexity when $k$ is constant?
• RQ3 How does the proposed method scale in high-dimensional Euclidean and metric spaces?
• RQ4 Can the approximation factor be improved for $k$-center clustering beyond the prior state-of-the-art?
• RQ5 What is the trade-off between approximation quality and runtime in the proposed framework?

Key findings

• The paper presents a $4$-approximation algorithm for balanced $k$-center clustering, improving upon the previous best approximation factor.
• For balanced $k$-median and $k$-means, the method achieves $O(1)$-approximation in $O(nd + (\log\Delta)^{3k}\log^2(\log\Delta))$ time when $k$ is constant.
• With refined candidate generation from [14], the algorithm yields a $(1+\epsilon)$-approximation for $k$-median and $k$-means in $O(2^{\tilde{O}(k/\epsilon^{O(1)})}(nd + (\frac{\log(\Delta/\epsilon)}{\epsilon})^{3k}(\log\frac{1}{\epsilon} + \log\log\Delta)^2))$ time.
• The framework is extendable to metric spaces, achieving $O(1)$-approximation in $O(nD + (\log\Delta)^{3k}\log^2(\log\Delta))$ time for metric $k$-BMedian and $k$-BMeans.
• The running time is nearly linear in $n$ for fixed $k$, significantly improving upon prior methods that relied on expensive flow or matching computations.
• Theoretical analysis confirms that the method maintains approximation guarantees through LP relaxation and rounding, with bounds on $r_{\text{max}}/r_{\text{min}}$ controlled via spread ratio and $\epsilon$-sampling.