On r-Gatherings on the Line

Abstract

In this paper we study a recently proposed variant of the facility location problem, called the $r$-gathering problem. Given an integer $r$, a set $C$ of customers, a set $F$ of facilities, and a connecting cost $co(c,f)$ for each pair of $c\in C$ and $f\in F$, an $r$-gathering of customers $C$ to facilities $F$ is an assignment $A$ of $C$ to open facilities $F^{'} \subset F$ such that $r$ or more customers are assigned to each open facility. We give an algorithm to find an $r$-gathering with the minimum cost, where the cost is $\max _{c_i\in C}\{co(c_i,A(c_i))\}$, when all $C$ and $F$ are on the real line.