This paper is divided into two parts. In the first part of this paper, we present a 2-approximation algorithm for the soft-capacitated facility location problem. This achieves the integrality gap of the natural LP relaxation of the problem. The algorithm is based on an improved analysis of an algorithm for the linear facility location problem, and a bi-factor approximate-reduction from this problem to the soft-capacitated facility location problem. We will define and use the concept of bifac-tor approximate reductions to improve the approximation factor of several other variants of the facility location problem. In the second part of the paper, we present an alternative analysis of the authors' 1.52-approximation algorithm for the uncapacitated facility location problem, using a single factor-revealing LP. This answers an open question of [16]. Furthermore, this analysis, combined with a recent result of Tho-rup [21] shows that our algorithm can be implemented in quasi-linear time, achieving the best known approximation factor in the best possible running time.
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