[Paper Review] Capacitated Center Problems with Two-Sided Bounds and Outliers

Hu Ding, Lunjia Hu+2 more|arXiv (Cornell University)|Feb 24, 2017

Advanced Graph Theory Research23 references4 citations

TL;DR

This paper presents the first constant-factor approximation algorithms for capacitated $k$-center problems with both lower and upper capacity bounds, including outlier handling. It introduces a simple combinatorial algorithm that achieves better approximation factors than prior LP-based methods, particularly for the unconstrained center count case, while also providing improved bounds for the supplier and outlier variants.

ABSTRACT

In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-center problems with one-sided bounds (upper or lower) have been well studied in previous work, and a constant factor approximation was obtained. We are the first to study the capacitated center problem with both capacity lower and upper bounds (with or without outliers). We assume each vertex has a uniform lower bound and a non-uniform upper bound. For the case of opening exactly $k$ centers, we note that a generalization of a recent LP approach can achieve constant factor approximation algorithms for our problems. Our main contribution is a simple combinatorial algorithm for the case where there is no cardinality constraint on the number of open centers. Our combinatorial algorithm is simpler and achieves better constant approximation factor compared to the LP approach.

Motivation & Objective

• Address the gap in approximation algorithms for $k$-center problems with both lower and upper capacity constraints.
• Study the problem under realistic constraints: uniform lower bounds and non-uniform upper bounds on cluster sizes.
• Extend the framework to include outlier vertices that may be excluded from clustering.
• Provide improved approximation guarantees over existing LP-based approaches, especially for unconstrained center count.
• Generalize results to the $k$-supplier setting with capacity bounds and outliers.

Proposed method

• Propose a novel combinatorial algorithm for the unconstrained $k$-center problem with both lower and upper bounds, avoiding complex LP rounding.
• Use a tree decomposition approach via a skeleton data structure to reduce the problem to a tree instance, enabling recursive construction.
• Define integral local distance-2 transfers to model feasible center selections while respecting capacity bounds.
• Apply case analysis based on the capacity of the root node relative to its children’s upper bounds to ensure feasibility and approximation.
• Modify upper bounds in the case $U_r > U_{u^*}$ to reduce to a simpler case, preserving approximation guarantees.
• Leverage prior results on skeleton construction and distance-24 transfers for outlier handling, adapting them to the case with lower bounds.

Experimental results

Research questions

• RQ1 Can a constant-factor approximation be achieved for the $k$-center problem with both lower and upper capacity bounds?
• RQ2 How does the inclusion of outliers affect the approximability of capacitated center problems with two-sided bounds?
• RQ3 Can a combinatorial algorithm outperform existing LP-based methods in terms of approximation ratio and simplicity?
• RQ4 What is the impact of non-uniform upper bounds and uniform lower bounds on the design of approximation algorithms?
• RQ5 Can the framework be extended to the $k$-supplier problem with two-sided bounds and outliers?

Key findings

• A polynomial-time 9-approximation algorithm is achieved for the (L, {U_u}, k)-Center problem with non-uniform upper bounds.
• For the uniform upper bound case, a 6-approximation is obtained, improving over prior LP-based results.
• A 13-approximation is achieved for the (L, {U_u}, k)-Supplier problem, with a 9-approximation for the uniform case.
• For the outlier version, a 25-approximation is achieved for (L, {U_u}, k, p)-Center, and a 23-approximation for the uniform case.
• The soft-capacity variant achieves a 13-approximation for both (L, {U_u}, soft-k, p)-Center and (L, U, soft-k, p)-Center.
• The proposed combinatorial algorithm achieves better approximation factors than the LP-based approach, particularly in the unconstrained center count setting.