[Paper Review] An improved approximation algorithm for k-median problem using a new factor-revealing LP

Chenchen Wu, Dachuan Xu+2 more|arXiv (Cornell University)|Oct 15, 2014

Facility Location and Emergency Management11 references3 citations

TL;DR

This paper presents an improved approximation algorithm for the k-median problem with a ratio of 2.592 + ε, achieved through a novel (1, 1.93910094) bi-factor approximation for the facility location problem (FLP). The approach leverages a new factor-revealing linear program to tighten the analysis and outperform the prior best ratio of 2.611 + ε.

ABSTRACT

The k-median problem is a well-known strongly NP-hard combinatorial optimization problem of both theoretical and practical significance. The previous best approximation ratio for this problem is 2.611+ε(Bryka et al. 2014) based on an (1, 1.95238219) bi-factor approximation algorithm for the classical facility location problem (FLP). This work offers an improved algorithm with an approximation ratio 2.592 +εbased on a new (1, 1.93910094) bi-factor approximation algorithm for the FLP.

Motivation & Objective

• To improve the approximation ratio for the NP-hard k-median problem.
• To develop a tighter analysis of the facility location problem (FLP) using a new factor-revealing linear program.
• To achieve a better bi-factor approximation for FLP, specifically (1, 1.93910094), to enhance k-median performance.
• To reduce the approximation ratio below the previous best of 2.611 + ε.

Proposed method

• Proposes a new factor-revealing linear program to analyze the performance of the k-median algorithm.
• Designs a (1, 1.93910094) bi-factor approximation algorithm for the facility location problem (FLP).
• Uses the improved FLP approximation as a subroutine in the k-median algorithm.
• Applies the new factor-revealing LP to derive tighter bounds on the approximation ratio.
• Optimizes the trade-off between facility opening cost and assignment cost in the FLP framework.
• Employs a refined analysis technique to achieve a better overall approximation ratio for k-median.

Experimental results

Research questions

• RQ1 Can the approximation ratio for the k-median problem be improved beyond 2.611 + ε?
• RQ2 What is the tightest possible bi-factor approximation for the facility location problem (FLP) achievable via a new factor-revealing LP?
• RQ3 How does a tighter FLP approximation translate into improved k-median performance?
• RQ4 Can a new factor-revealing LP model yield better bounds than existing approaches in k-median approximation?
• RQ5 What is the minimal upper bound on the approximation ratio for k-median using this new analytical framework?

Key findings

• The proposed algorithm achieves an approximation ratio of 2.592 + ε for the k-median problem, improving upon the prior best ratio of 2.611 + ε.
• The new factor-revealing LP enables a tighter analysis, resulting in a (1, 1.93910094) bi-factor approximation for the facility location problem (FLP).
• The improvement in the FLP approximation directly contributes to the enhanced k-median performance.
• The new approach reduces the approximation ratio by approximately 0.019, representing a significant theoretical advancement.
• The method demonstrates that refined factor-revealing LP models can yield better bounds in combinatorial optimization.
• The result confirms the effectiveness of the new LP formulation in tightening approximation guarantees.