A Concentration Inequality for the Facility Location Problem

TL;DR

This paper establishes a concentration inequality for a stochastic facility location problem, showing the objective's values are tightly concentrated around its expectation for uniform random points in the unit square.

Contribution

It introduces a novel concentration inequality for the facility location problem using geometric and martingale techniques, extending previous approximation methods.

Findings

Objective $C_n$ concentrates in an $O(n^{1/6})$ interval.

Expected $C_n$ scales as $ heta(n^{2/3})$ for uniform points.

Techniques generalize to other stochastic geometric settings.

Abstract

We give a concentration inequality for a stochastic version of the facility location problem. We show the objective $C_n = \min_{F \subseteq [0,1]^2}|F|+\sum_{x\in X}\min_{f\in F}\|x-f\|$ 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.