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 didnt 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.
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