FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

TL;DR

This work studies Capacitated Sum of Radii, where each potential center has a hard capacity and one must choose $k$ centers to minimize the sum of radii. It delivers the first fixed-parameter tractable constant-factor approximations, including a $(15+ε)$-approximation for non-uniform capacities in general metrics and several sharp uniform-capacity results, with running times that scale as $2^{\mathcal{O}(k^2\log k)}\cdot n^3$ or better in various settings; in Euclidean space, $(1+ε)$- and $(2+ε)$-approximations are achieved with dimension-dependent time bounds. The authors also introduce bi-criteria relaxations that permit limited capacity violations to obtain tighter cost guarantees, along with dimension-aware approximation schemes. Comprehensive hardness results (ETH-based and NP-hardness) show that these results are near-tight, especially in general metrics, and even stronger limits hold in the Euclidean setting. Overall, the paper advances the understanding of capacitated clustering by providing the first substantial FPT progress and establishing clear barriers for polynomial-time constant-factor approaches, while offering novel techniques such as large-small ball decompositions and random sampling strategies.

Abstract

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into $k$ clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a $(15+ε)$-approximation algorithm that runs in $2^{0(k^2\log k)}\cdot n^3$ time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + $ε$)-approximation with running time $2^{O(k\log(k/ε))}n^3$, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + $ε$)-approximation with running time $2^{O(k/ε^2 \cdot\log(k/ε))}dn^3$ and a $(1+ε)$-approximation with running time $2^{O(kd\log ((k/ε)))}n^{3}$ in the Euclidean space; and a (1 + $ε$)-approximation in the Euclidean space with running time $2^{O(k/ε^2 \cdot\log(k/ε))}dn^3$ if we are allowed to violate the capacities by (1 + $ε$)-factor. We complement this result by showing that there is no (1 + $ε$)-approximation algorithm running in time $f(k)\cdot n^{O(1)}$, if any capacity violation is not allowed.

Figures (2)

• Figure 1: Figure showing construction of $L_1\cup L_2$. $G^\star$ has five components: $C_1=\{B^\star_1,B^\star_2\}$, $C_2=\{B^\star_3\}$, $C_3=\{B^\star_4\}$, $C_4=\{B^\star_5\}$. Also, $\mathcal{B}^\star_1=\{B^\star_1,B^\star_5\}$ and $\mathcal{B}^\star_2=\{B^\star_2,B^\star_3,B^\star_4\}$. For this example, $L_1=\{B_1,B_5\}$ and $L_2=\{B_{C_1}\}$.
• Figure 2: Figure showing construction of $L_3$ and $L_4$ for the example in Figure \ref{['fig:fig1']}. Here $L_3=\{B_3\}$ and $L_4=\{T_5\}$.

Theorems & Definitions (73)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1
• Lemma 2.1