A concentration inequality for the facility location problem

We give a concentration inequality for a stochastic version of the facility location problem. We show the objective C-n = min(F subset of[0, 1]2) vertical bar F vertical bar + Sigma(x is an element of X )min(f is an element of F) parallel to x - f parallel to is concentrated in an interval of length O (n(1/6)) and E[C-n] = Theta(n(2/3)) if the input X consists of i.i.d. uniform points in the unit square. Our main tool is to use a geometric quantity, previously used in the design of approximation algorithms for the facility location problem, to analyze a martingale process. Many of our techniques generalize to other settings. (C) 2022 Elsevier B.V. All rights reserved.