Constrained k-Center Problem on a Convex Polygon

Abstract

In this paper, we consider a restricted covering problem, in which a convex polygon ${\mathcal{P}}$ with n vertices and an integer k are given, the objective is to cover the entire region of ${\mathcal{P}}$ using k congruent disks of minimum radius $r_{opt}$, centered on the boundary of ${\mathcal{P}}$. For ${k\ge 7}$ and any ${\epsilon >0}$, we propose a ${(1+\frac{7}{k}+\frac{7\epsilon }{k}+\epsilon )}$-factor approximation algorithm, which runs in ${O(n(n+k)(|{\log r_{opt}}|+\log \lceil \frac{1}{\epsilon }\rceil ))}$ time. The previous best known approximation factor in the literature for the same problem is 1.8841 [H. Du and Y. Xu: An approximation algorithm for k-center problem on a convex polygon, J. Comb. Optim. (2014), 27(3), 504-518].