A Concentration Inequality for the Facility Location Problem

12/08/2020

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by   Sandeep Silwal, et al.

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We give a concentration inequality for a stochastic version of the facility location problem on the plane. We show the objective C_n(X) = min_F ⊆ [0,1]^2 |F| + ∑_x∈ Xmin_f ∈ F x-f is concentrated in an interval of length O(n^1/6) and 𝔼[C_n] = Θ(n^2/3) if the input X consists of n i.i.d. uniform points in the unit square. Our main tool is to use a suitable geometric quantity, previously used in the design of approximation algorithms for the facility location problem, to analyze a martingale process.