On min-max r-gatherings

Amitai Armon

Abstract: We consider a min–max version of the previously studied r-gathering problem with unit-demands. The problem we consider is a metric facility-location problem, in which each open facility must serve at least r customers, and the maximum of all the facility- and connection-costs should be minimized (rather than their sum ). This problem is motivated by scenarios in which r customers are required for a facility to be worth opening, and the costs represent the time until the facility/connection will be available ( i.e. , we want to have the complete solution ready as soon as possible). We present a 3-approximation algorithm for this problem, and prove that it cannot be approximated better (assuming P ≠ N P ). Next we consider this problem with the additional natural requirement that each customer will be assigned to a nearest open facility, and present a 9-approximation algorithm. We further consider previously introduced special cases and variants, and obtain improved algorithmic and hardness results.