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We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61. We use this algorithm to find better approximation algorithms for the capacitated facility location problem with soft capacities and for a common generalization of the k-median and facility location problems. We also prove a lower bound of 1+2/e on the approximability of the k-median problem. At the end, we present a discussion about the techniques we have used in the analysis of our algorithm, including a computer-aided method for proving bounds on the approximation factor.