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Coresets for Constrained Clustering: General Assignment Constraints and Improved Size Bounds

Lingxiao Huang, Jian Li, Pinyan Lu, Xuan Wu

TL;DR

A general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body $\mathcal{B}$) are introduced.

Abstract

Designing small-sized \emph{coresets}, which approximately preserve the costs of the solutions for large datasets, has been an important research direction for the past decade. We consider coreset construction for a variety of general constrained clustering problems. We introduce a general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body $\mathcal{B}$). We give coresets for clustering problems with such general assignment constraints that significantly generalize and improve known results. Notable implications include the first $\varepsilon$-coreset for capacitated and fair $k$-Median with $m$ outliers in Euclidean spaces whose size is $\tilde{O}(m + k^2 \varepsilon^{-4})$, generalizing and improving upon the prior bounds in [Braverman et al., FOCS' 22; Huang et al., ICLR' 23] (for capacitated $k$-Median, the coreset size bound obtained in [Braverman et al., FOCS' 22] is $\tilde{O}(k^3 \varepsilon^{-6})$, and for $k$-Median with $m$ outliers, the coreset size bound obtained in [Huang et al., ICLR' 23]} is $\tilde{O}(m + k^3 \varepsilon^{-5})$), and the first $ε$-coreset of size $\mathrm{poly}(k \varepsilon^{-1})$ for fault-tolerant clustering for various types of metric spaces.

Coresets for Constrained Clustering: General Assignment Constraints and Improved Size Bounds

TL;DR

A general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body ) are introduced.

Abstract

Designing small-sized \emph{coresets}, which approximately preserve the costs of the solutions for large datasets, has been an important research direction for the past decade. We consider coreset construction for a variety of general constrained clustering problems. We introduce a general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body ). We give coresets for clustering problems with such general assignment constraints that significantly generalize and improve known results. Notable implications include the first -coreset for capacitated and fair -Median with outliers in Euclidean spaces whose size is , generalizing and improving upon the prior bounds in [Braverman et al., FOCS' 22; Huang et al., ICLR' 23] (for capacitated -Median, the coreset size bound obtained in [Braverman et al., FOCS' 22] is , and for -Median with outliers, the coreset size bound obtained in [Huang et al., ICLR' 23]} is ), and the first -coreset of size for fault-tolerant clustering for various types of metric spaces.
Paper Structure (56 sections, 35 theorems, 191 equations, 1 table, 1 algorithm)

This paper contains 56 sections, 35 theorems, 191 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. General Assignment Constraints
  3. Coresets
  4. Coreset for Constrained Clustering
  5. Our Contributions
  6. A General Model of Assignment Constraints
  7. Main Theorem
  8. New Coreset Results and Improved Bounds
  9. Upper Bound of $\mathsf{Lip}(\mathcal{B})$ when $\mathcal{B}$ is Matroid Basis Polytope
  10. Related Work
  11. Technical Overview
  12. Size Improvement
  13. Handling Assignment Structure Constraints
  14. Modeling and Definitions
  15. General Assignment Constraints
  16. ...and 41 more sections

Key Result

Theorem 1.1

We consider $(k, z)$-Clustering with capacity upper/lower bound constraint for each center, assignment structure constraint for each point (specified by convex body $\mathcal{B} \subseteq \Delta_k$), and a total capacity constraint $\|\sigma\|_1=n-m$ (i.e., $m$ outliers). For any $0 < \varepsilon <

Theorems & Definitions (108)

  • Theorem 1.1: Informal; see Theorem \ref{['thm:coreset']}
  • Definition 2.1: Capacity constraint
  • Definition 2.2: Total capacity constraint
  • Definition 2.3: Assignment structure constraint
  • Definition 2.4: Coreset
  • Claim 2.5: Capacitated Clustering
  • proof
  • Claim 2.6: Fair Clustering
  • proof
  • Definition 3.1: $(\alpha,\beta,\gamma)$-Approximation for $(k, z)$-Clustering with $m$ outliers
  • ...and 98 more