Before revision:  Previously, the only known solution for this problem with a sub-linear competitive ratio was randomized Alon et al. , TALG 2006].
After revision:  Our algorithm improves over the O((\log |C| + \log |F|) \cdot (\log |C| + \log \log |F|))-competitive construction that first reduces the facility location instance to a set cover one and then later solves such instance using the deterministic algorithm by Alon et al. , TALG 2006].