Squared Metric Facility Location Problem

Jain et al. proposed two well-known algorithms for the Metric Facility Location Problem (MFLP), that achieve approximation ratios of 1.861 and 1.61. Mahdian et al. combined the latter algorithm with scaling and greedy augmentation techniques, obtaining a 1.52-approximation for the MFLP. We consider a generalization of the Squared Euclidean Facility Location Problem, when the distance function is a squared metric, which we call Squared Metric Facility Location Problem (SMFLP). We show that the algorithms of Jain et al. and of Mahdian et al., when applied to this variant of the facility location, achieve approximation ratios of 2.87, 2.43, and 2.17, respectively. It is shown that, for the SMFLP, there is no 2.04-approximation algorithm, assuming P = NP. In our analysis, we used nonlinear factor-revealing programs to obtain both lower and upper bounds on the approximation factors, and propose a systematic way to derive such factor-revealing programs.