Approximation algorithms for finding maximum containing circle and sphere

We first study maximum containing circle problem. The input to the problem is a weighted set of points and a circle of fixed radius, and the output is a suitable location of the circle such that the sum of the weights of the points covered by the circle is maximized. We propose a special polygon, called symmetrical rectilinear polygon (SRP). In this paper, we give a method for constructing the circumscribed SRP of a circle and prove the area relationship between this polygon and the circle. We solve the maximum containing SRP problem exactly, and based on this, give an algorithm for solving the $(1-\epsilon )$-approximation of maximum containing circle problem. We also show that the algorithm is valid for most inputs. It only needs $O(n{\epsilon }^{-1}\log n+n{\epsilon }^{-1}\log \left(\frac{1}{\epsilon }\right))$ time for unit points and $o\left(n{\epsilon }^{-2}\log n\right)$ time for weighted points. Due to its low time complexity, the algorithm can run as a stand-alone algorithm or as a preprocessor for other algorithms. As an extension of our work, we discuss a 3D version of the unweighted maximum containing circle problem, i.e., containing the maximum number of points with a given sphere. We give a $(1-\epsilon )$-approximation algorithm for this problem that returns correct results in $max\{O\left((n^{\frac{3}{2}}{\epsilon }^{-3}/{\log }^{\frac{3}{2}}(n{\epsilon }^{-2}\left)\right){(\log \log (n{\epsilon }^{-2}\left)\right)}^{O\left(1\right)}\right),o\left(n^2{\left({\epsilon }^{-1}\log n\right)}^2\right)\}$ time for most cases.