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A Distributed Approximation Algorithm for the Metric Uncapacitated Facility Location Problem in the Congest Model

We present a randomized distributed approximation algorithm for the metric uncapacitated facility location problem. The algorithm is executed on a bipartite graph in the Congest model yielding a (1.861 + epsilon) approximation factor, where epsilon is an arbitrary small positive constant. It needs O(n^{3/4}log_{1+epsilon}^2(n) communication rounds with high probability (n denoting the number of facilities and clients). To the best of our knowledge, our algorithm currently has the best approximation factor for the facility location problem in a distributed setting. It is based on a greedy sequential approximation algorithm by Jain et al. (J. ACM 50(6), pages: 795-824, 2003). The main difficulty in executing this sequential algorithm lies in dealing with situations, where multiple facilities are eligible for opening, but (in order to preserve the approximation factor of the sequential algorithm) only a subset of them can actually be opened. Note that while the presented runtime bound of our algorithm is "with high probability", the approximation factor is not "in expectation" but always guaranteed to be (1.861 + epsilon). Thus, our main contribution is a sublinear time selection mechanism that, while increasing the approximation factor by an arbitrary small additive term, allows us to decide which of the eligible facilities to open.