An Attempt to Design a Better Algorithm for the Uncapacitated Facility Location Problem

TL;DR

This paper reviews existing algorithms for the uncapacitated facility location problem, offering insights and a new vector game framework to guide the design of balanced algorithms, but does not improve current approximation ratios.

Contribution

It introduces a vector game framework for the problem, providing new insights and a basis for designing balanced algorithms, despite not surpassing existing approximation ratios.

Findings

Reviewed and analyzed existing algorithms and their approximation ratios.

Proposed a vector game framework to approach the problem.

Provided insights that may guide future algorithm design.

Abstract

The uncapacitated facility location has always been an important problem due to its connection to operational research and infrastructure planning. Byrka obtained an algorithm that is parametrized by $\gamma$ and proved that it is optimal when $\gamma>1.6774$. He also proved that the algorithm achieved an approximation ratio of 1.50. A later work by Shi Li achieved an approximation factor of 1.488. In this research, we studied these algorithms and several related works. Although we didn't improve upon the algorithm of Shi Li, our work did provide some insight into the problem. We also reframed the problem as a vector game, which provided a framework to design balanced algorithms for this problem.