An incremental version of the k-center problem on boundary of a convex polygon

Abstract

This paper studies an incremental version of the k-center problem with centers constrained to lie on boundary of a convex polygon. In the incremental k-center problem we considered, we are given a set of n demand points inside a convex polygon, facilities are constrained to lie on its boundary. Our algorithm produces an incremental sequence of facility sets $B_{1}\subseteq B_{2}\subseteq \cdots \subseteq B_{n}$, where each $B_{k}$ contains k facilities. Such an algorithm is called $\alpha$-competitive, if for any k, the maximum of the ratio between the value of solution $B_{k}$ and the value of an optimal k-center solution is no more than $\alpha$. We present a polynomial time incremental algorithm with a competitive ratio $\frac{3\sqrt{3}}{2}$ and we also prove a lower bound of 2.