[Paper Review] FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

Sayan Bandyapadhyay, William Lochet+1 more|arXiv (Cornell University)|Jan 1, 2023

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TL;DR

This paper presents the first fixed-parameter tractable (FPT) constant-factor approximation algorithm for the capacitated sum of radii clustering problem, achieving a (15 + ϵ)-approximation in time 2^O(k² log k) · n³. It further improves approximation ratios to (4 + ϵ) and (2 + ϵ) for uniform capacities in general and Euclidean metrics, respectively, and provides (1 + ϵ)-approximations with capacity violations or in FPT time, while proving that (1 + ϵ)-approximation without capacity violation is impossible in FPT time under standard assumptions.

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.

Motivation & Objective

• To address the long-standing open question of whether a polynomial-time constant-approximation exists for the capacitated sum of radii clustering problem.
• To develop fixed-parameter tractable (FPT) algorithms that achieve constant-factor approximations despite the hardness introduced by capacity constraints.
• To explore the limits of approximation in both general and Euclidean metrics, particularly under capacity violations or FPT time constraints.
• To establish tight hardness results showing that (1 + ϵ)-approximation without capacity violation is impossible in FPT time.

Proposed method

• Designs a novel FPT algorithm using iterative rounding and clustering decomposition techniques to handle non-uniform capacities.
• Applies a primal-dual framework combined with Lagrangean relaxation to manage capacity constraints in the sum of radii objective.
• Introduces a coreset-based approach for Euclidean spaces to achieve (1 + ϵ)-approximations with capacity violations.
• Employs a reduction from the k-clique problem to prove inapproximability results, showing that (1 + ϵ)-approximation without capacity violation is impossible in FPT time.
• Uses geometric constructions in Euclidean space to establish lower bounds on the cost gap between yes and no instances of k-clique.
• Leverages the Johnson Coverage Hypothesis and existing hardness results to strengthen the inapproximability argument.

Experimental results

Research questions

• RQ1 Does the capacitated sum of radii problem admit a polynomial-time constant-factor approximation, even with uniform capacities?
• RQ2 What is the best possible approximation ratio achievable in FPT time for the capacitated sum of radii problem?
• RQ3 Can (1 + ϵ)-approximation be achieved in FPT time for the Euclidean version of the problem without violating capacity constraints?
• RQ4 Is there a fundamental barrier preventing (1 + ϵ)-approximation in FPT time when no capacity violation is allowed?
• RQ5 What is the trade-off between approximation ratio, running time, and capacity violation in Euclidean and general metrics?

Key findings

• The paper presents the first FPT constant-factor (15 + ϵ)-approximation algorithm for the non-uniform capacitated sum of radii problem, running in 2^O(k² log k) · n³ time.
• For uniform capacities, a (4 + ϵ)-approximation is achieved in 2^O(k log(k/ϵ)) · n³ time, significantly improving over the prior FPT 28-approximation.
• In Euclidean space, a (2 + ϵ)-approximation is obtained in 2^O(k/ϵ² · log(k/ϵ)) · d·n³ time, and a (1 + ϵ)-approximation in 2^O(kd log((k/ϵ))) · n³ time.
• A (1 + ϵ)-approximation is possible in Euclidean space with (1 + ϵ)-capacity violation, running in 2^O(k/ϵ² · log(k/ϵ)) · d·n³ time.
• The paper proves that no (1 + ϵ)-approximation algorithm exists in f(k) · n^O(1) time without capacity violation, establishing a tight inapproximability barrier.
• The hardness result is based on a reduction from the k-clique problem, showing a multiplicative gap of (1 + 1/(12αn⁵)) between yes and no instances, implying that FPTAS is impossible unless P = NP.