Away from Each Other

Abstract

We consider the following three problems where k is a constant. For those problems there are cases where k is typically a small constant. Given a polygon with n edges on a plane we want to find k points in the polygon so that the minimum pairwise Euclidean distance of the k points is maximized. Intuitively, for an island, we want to locate k drone bases far away from each other in flying distance to avoid congestion in the sky. In this paper we give an $$O( ((1/\epsilon )^{2} + n/\epsilon )^k )$$ time $$1/(1+\epsilon )$$ approximation algorithm to solve the problem, where $$\epsilon < 1$$ is a positive number. This is the first PTAS for the problem. Given a set of n straight line segments on a plane we want to find k points on the straight line segments so that the minimum pairwise Euclidean distance of the k points is maximized. Intuitively, for some road network, we want to locate k drone bases far away from each other to avoid congestion in the sky and also each base face a road. In this paper we design the first PTAS for the problem. Given a polygon with n edges on a plane we want to find k points in the polygon so that the minimum length of paths inside the polygon connecting two points among the k points is maximized. Intuitively, for an island, we want to locate k coffee shops far away from each other to avoid self competition for walking customers. In this paper we design the first PTAS for the problem.