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Improved fixed-parameter bounds for Min-Sum-Radii and Diameters $k$-clustering and their fair variants

Sandip Banerjee, Yair Bartal, Lee-Ad Gottlieb, Alon Hovav

TL;DR

The paper advances the algorithmic theory of MSR and MSD clustering by delivering tight fixed-parameter bounds: it shows an exact $n^{O(k)}$-time MSD algorithm (matching the known $n^{O(k)}$ bound for MSR) and ETH-based lower bounds, while providing $(1+\epsilon)$-approximation schemes in metrics of doubling dimension $d$ with run-time $O(kn) + (1/\epsilon)^{O(dk)}$ (and $\left(\alpha/\epsilon\right)^{O(dk)}$ for $\alpha$-MSR). These results extend to fairness and mergeable clustering variants, as well as to outliers, preserving the same asymptotic guarantees, with additional combinatorial factors where necessary. A decomposition-based framework underpins the approximations, leveraging nets and bounded-diameter components to achieve deterministic PTAS-like performance in low-dimension spaces. ETH-based hardness results establish tight barriers for several variants, clarifying when efficient FPT algorithms are possible. Overall, the work significantly strengthens the practical and theoretical foundations for fixed-parameter clustering in general metric spaces.

Abstract

We provide improved upper and lower bounds for the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems with a bounded number of clusters $k$. In particular, we propose an exact MSD algorithm with running-time $n^{O(k)}$. We also provide $(1+ε)$ approximation algorithms for both MSR and MSD with running-times of $O(kn) +(1/ε)^{O(dk)}$ in metrics spaces of doubling dimension $d$. Our algorithms extend to $k$-center, improving upon previous results, and to $α$-MSR, where radii are raised to the $α$ power for $α>1$. For $α$-MSD we prove an exponential time ETH-based lower bound for $α>\log 3$. All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe fairness constraints, as well as to the general framework of mergeable clustering, which includes many other popular clustering variants. We complement these upper bounds with ETH-based lower bounds for these problems, in particular proving that $n^{O(k)}$ time is tight for MSR and $α$-MSR even in doubling spaces, and that $2^{o(k)}$ bounds are impossible for MSD.

Improved fixed-parameter bounds for Min-Sum-Radii and Diameters $k$-clustering and their fair variants

TL;DR

The paper advances the algorithmic theory of MSR and MSD clustering by delivering tight fixed-parameter bounds: it shows an exact -time MSD algorithm (matching the known bound for MSR) and ETH-based lower bounds, while providing -approximation schemes in metrics of doubling dimension with run-time (and for -MSR). These results extend to fairness and mergeable clustering variants, as well as to outliers, preserving the same asymptotic guarantees, with additional combinatorial factors where necessary. A decomposition-based framework underpins the approximations, leveraging nets and bounded-diameter components to achieve deterministic PTAS-like performance in low-dimension spaces. ETH-based hardness results establish tight barriers for several variants, clarifying when efficient FPT algorithms are possible. Overall, the work significantly strengthens the practical and theoretical foundations for fixed-parameter clustering in general metric spaces.

Abstract

We provide improved upper and lower bounds for the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems with a bounded number of clusters . In particular, we propose an exact MSD algorithm with running-time . We also provide approximation algorithms for both MSR and MSD with running-times of in metrics spaces of doubling dimension . Our algorithms extend to -center, improving upon previous results, and to -MSR, where radii are raised to the power for . For -MSD we prove an exponential time ETH-based lower bound for . All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe fairness constraints, as well as to the general framework of mergeable clustering, which includes many other popular clustering variants. We complement these upper bounds with ETH-based lower bounds for these problems, in particular proving that time is tight for MSR and -MSR even in doubling spaces, and that bounds are impossible for MSD.

Paper Structure

This paper contains 41 sections, 50 theorems, 1 figure, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Exact algorithms.
  3. Approximation algorithms.
  4. Extension to fairness and mergeable clustering.
  5. Related work
  6. MSR:
  7. MSD:
  8. Preliminaries and definitions
  9. Definitions and notation.
  10. Nets and point hierarchies.
  11. Min-sum radii clustering.
  12. Fair-MSR
  13. Min-sum diameters clustering.
  14. Mergeable min-sum diameters clustering.
  15. Clustering with outliers.
  16. ...and 26 more sections

Key Result

Proposition 1

The MSR problem and the $\alpha$-MSR problem with $k$ clusters can be solved exactly in time $n^{O(k)}$ in general metric spaces. This holds also in the case of $g$ outliers.

Figures (1)

  • Figure 1: MSR Hardness: In the drawing, we see an example for the reduction, with some of the distances drawn. In this example, the pairs $s_i^1 \in S_{i,i}$ and $s_j^1 \in S_{j,j}$, corresponding to $t_i^1,t_j^1$ respectively, are feasible.

Theorems & Definitions (81)

  • Proposition 1
  • Definition 1: Neighborhood
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Definition 2: Decomposability
  • Lemma 5
  • Definition 3
  • ...and 71 more