Clustering Geometrically-Modeled Points in the Aggregated Uncertainty Model
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The k-center problem is to choose a subset of size k from a set of n points such that the maximum distance from each point to its nearest center is minimized. Let Q={Q_1,…,Q_n} be a set of polygons or segments in the region-based uncertainty model, in which each Q_i is an uncertain point, where the exact locations of the points in Q_i are unknown. The geometric objects segments and polygons can be models of a point set. We define the uncertain version of the k-center problem as a generalization in which the objective is to find k points from Q to cover the remaining regions of Q with minimum or maximum radius of the cluster to cover at least one or all exact instances of each Q_i, respectively. We modify the region-based model to allow multiple points to be chosen from a region and call the resulting model the aggregated uncertainty model. All these problems contain the point version as a special case, so they are all NP-hard with a lower bound 1.822. We give approximation algorithms for uncertain k-center of a set of segments and polygons. We also have implemented some of our algorithms on a data-set to show our theoretical performance guarantees can be achieved in practice.
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