r-Gatherings on a Star

Abstract

Let C be a set of n customers and F be a set of m facilities. An r-gather clustering of C is a partition of the points in clusters such that each cluster contains at least r points. The r-gather clustering problem asks to find an r-gather clustering which minimizes the maximum distance between any two points in a cluster. An r-gathering of C is an assignment of each customer $c \in C$ to a facility $f \in F$ such that each open facility has zero or at least r customers. The r-gathering problem asks to find an r-gathering that minimizes the maximum distance between a customer and its facility. In this work we consider the r-gather clustering and r-gathering problems when the customers and the facilities are lying on a “star”. We show that the r-gather clustering problem and the r-gathering problem with points on a star with d rays can be solved in $O(rn+(r+1)^ddr)$ and $O(n+r^2 m + d^2r^2 (d+\log m)+(r+1)^d2^d(r +d)d)$ time respectively.