K-robots clustering of moving sensors using coresets

We present an approach to position k servers (e.g. mobile robots) to provide a service to n independently moving clients; for example, in mobile ad-hoc networking applications where inter-agent distances need to be minimized, connectivity constraints exist between servers, and no a priori knowledge of the clients´ motion can be assumed. Our primary contribution is an algorithm to compute and maintain a small representative set, called a kinematic coreset, of the n moving clients.We prove that, in any given moment, the maximum distance between the clients and any set of k servers is approximated by the coreset up to a factor of (1 ± ε), where ε > 0 is an arbitrarily small constant. We prove that both the size of our coreset and its update time is polynomial in k log(n)/ε. Although our optimization problem is NP-hard (i.e., takes time exponential in the number of servers to solve), solving it on the small coreset instead of the original clients results in a tractable controller. The approach is validated in a small scale hardware experiment using robot servers and human clients, and in a large scale numerical simulation using thousands of clients.