r-Gatherings on a star and uncertain r-gatherings on a line

Abstract

Let [Formula: see text] be a set of [Formula: see text] customers and [Formula: see text] be a set of [Formula: see text] facilities. An [Formula: see text]-gather clustering of [Formula: see text] is a partition of the customers in clusters such that each cluster contains at least [Formula: see text] customers. The [Formula: see text]-gather clustering problem asks to find an [Formula: see text]-gather clustering which minimizes the maximum distance between a pair of customers in a cluster. An [Formula: see text]-gathering of [Formula: see text] to [Formula: see text] is an assignment of each customer [Formula: see text] to a facility [Formula: see text] such that each facility has zero or at least [Formula: see text] customers. The [Formula: see text]-gathering problem asks to find an [Formula: see text]-gathering that minimizes the maximum distance between a customer and his/her facility. In this work, we consider the [Formula: see text]-gather clustering and [Formula: see text]-gathering problems when the customers and the facilities are lying on a “star”. We show that the [Formula: see text]-gather clustering problem and the [Formula: see text]-gathering problem with customers and facilities on a star with [Formula: see text] rays can be solved in [Formula: see text] and [Formula: see text] time, respectively. Furthermore, we prove the hardness of a variant of the [Formula: see text]-gathering problem, called the min-max-sum [Formula: see text]-gathering problem, even when the customers and the facilities are on a star. We also study the [Formula: see text]-gathering problem when the customers and the facilities are on a line, and each customer location is uncertain. We show that the [Formula: see text]-gathering problem can be solved in [Formula: see text] and [Formula: see text] time when the customers and the facilities are on a line, and the customer locations are given by piecewise uniform functions of at most [Formula: see text] pieces and “well-separated” uniform distribution functions, respectively.