The p-center problem under locational uncertainty of demand points

The p-center problem is finding the location of p facilities among a set of n demand points such that the maximum distance between any demand point and its nearest facility is minimized. In this paper, we study this problem in the context of uncertainty, that is, the location of the demand points may change in a region like a disk or a segment, or belong to a finite set of points. We introduce Max-p-center and Min-p-center problems which are the worst and the best possible solutions for the p-center problem under such locational uncertainty. We propose approximation and parameterized algorithms to solve these problems under the Euclidean metric. Further, we study the MinMax Regret 1-center problem under uncertainty and propose a linear-time algorithm to solve it under the Manhattan metric as well as an O(n4) time algorithm under the Euclidean metric.