A Simple Algorithm for r-gatherings on the Line

Abstract

In this paper we study recently proposed variant of the facility location problem called the r-gathering problem. Given sets C and F of points on the plane and distance d(c, f) for each $c\in C$ and $f\in F$, an r-gathering of C to 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. The cost of an r-gathering is the maximum distance d(c, f) between $c\in C$ and $A(c)\in F'$ among the assignment, which is $\max _{c\in C}\{ d(c,A(c)) \}$. The r-gathering problem finds the r-gathering minimize the cost. A polynomial time 3-approximation algorithm for the r-gathering problem is known. When all C and F are on the line an $O((|C|+|F|)\log (|C|+|F|) )$ time algorithm and an $O(|C|+|F|\log ^2 r+|F|\log |F|)$ time algorithm to solve the r-gathering problem are known. In this paper we give a simple $O(|C|+r^2|F|)$ time algorithm to solve the r-gathering problem. Since in typical case $r<<|F|<<|C|$ holds our new algorithm is faster than the known algorithms.