[Paper Review] On the Factor Revealing LP Approach for Facility Location with Penalties

Xian Qiu, Walter Kern|arXiv (Cornell University)|Jan 31, 2016

Facility Location and Emergency Management20 references3 citations

TL;DR

This paper presents a modified JMS algorithm combined with randomized LP rounding to achieve a 1.488 approximation ratio for the uncapacitated facility location problem with linear penalties, improving upon the prior 1.5148 bound. The key innovation lies in adapting the JMS algorithm to the penalty setting and using a monotone factor revealing LP technique, closing the gap between classical and penalized facility location variants.

ABSTRACT

We consider the uncapacitated facility location problem with (linear) penalty function and show that a modified JMS algorithm, combined with a randomized LP rounding technique due to Byrka-Aardal[1], Li[14] and Li et al.[16] yields 1.488 approximation, improving the factor 1.5148 due to Li et al.[16]. This closes the current gap between the classical facility location problem and this penalized variant. Main ingredient is a straightforward adaptation of the JMS algorithm to the penalty setting plus a consistent use of the upper bounding technique for factor revealing LPs due to Fernandes et al.[7]. In contrast to the bounds in [12], our factor revealing LP is monotone.

Motivation & Objective

• To address the uncapacitated facility location problem with linear penalty functions.
• To improve upon the existing 1.5148 approximation ratio for the penalized variant.
• To close the performance gap between the classical facility location problem and its penalized counterpart.
• To develop a monotone factor revealing LP framework suitable for the penalty setting.

Proposed method

• Adapting the JMS algorithm to the penalty-based facility location setting through a straightforward modification.
• Applying randomized LP rounding techniques from Byrka-Aardal, Li, and Li et al. to the modified algorithm.
• Employing the upper bounding technique for factor revealing LPs from Fernandes et al. to analyze approximation guarantees.
• Constructing a monotone factor revealing LP to ensure consistent and tight performance bounds.
• Using the monotonicity of the factor revealing LP to simplify and strengthen the analysis.

Experimental results

Research questions

• RQ1 Can the JMS algorithm be effectively adapted to the facility location problem with linear penalties?
• RQ2 What approximation ratio can be achieved using a modified JMS algorithm combined with randomized LP rounding in the penalized setting?
• RQ3 How does the performance of the new algorithm compare to the prior 1.5148 bound by Li et al.?
• RQ4 Can a monotone factor revealing LP be constructed for the penalized facility location problem to ensure tighter analysis?

Key findings

• The proposed algorithm achieves a 1.488 approximation ratio for the uncapacitated facility location problem with linear penalties.
• This improves upon the previous best-known approximation ratio of 1.5148 established by Li et al.
• The factor revealing LP used in the analysis is monotone, enabling a more robust and consistent performance guarantee.
• The adaptation of the JMS algorithm to the penalty setting is straightforward and effective.