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A Sub-quadratic Time Algorithm for the Proximity Connected k-center Problem on Paths via Modular Arithmetic

The k -center problem is one of the most well-known problems in combinatorial optimization which has been extensively studied in the past. In this paper, we introduce a generalized version of the k -center problem called proximity connected k -center (PCkC) problem. In this problem, we are given a set of demand points in a metric space and a parameter δ > 0. We are going to locate k center points such that the maximum distance of a demand point to its nearest center is minimized and each pair of centers can communicate with each other either directly or via other centers assuming that each center can directly communicate with any other center within the range of δ of itself. Note that when δ is large enough, the problem turns to the k -center problem and when δ tends to zero, the problem turns to the 1-center problem. We consider the PCkC problem when the underlying space is a path and present a sub-quadratic time algorithm for both the unweighted and the weighted demand points cases.