This note is a study on the problem of assigning non-overlapping disks centered at a set of given points in%DIFDELCMD < {\IR}%%% ^2 such that the sum of area covered by them is maximized. If the points are placed on a straight-line  , then the problem is solvable in polynomial time. However,  the problem is shown to be NP-hard  in { %DIFDELCMD < \IR%%%  }^2. Eppstein [CCCG, pages 260--265, 2016] proposed a polynomial time algorithm for maximizing the sum of radii (or perimeter) of non-overlapping balls or disks when the points are arbitrarily placed on a plane. We show that Eppstein's algorithm for maximizing sum of perimeter of the disks in { %DIFDELCMD < \IR%%%  }^2 gives a 2-approximation solution for the sum of area maximization problem. We propose a PTAS for our problem. These approximation results are extendible to higher dimension . All these approximation results hold for the area maximization problem by regular  polygons with even number of edges centered at the given points.
We study the problem of assigning non-overlapping geometric objects centered at a %DIFDELCMD < {\IR}%%% given set of points such that the sum of area covered by them is maximized. If the points are placed on a straight-line and the objects are disks , then the problem is solvable in polynomial time. However, we show that the problem is NP-hard even for simplest objects like disks or squares in { %DIFDELCMD < \IR%%% \mathbb{R }^2. Eppstein [CCCG, pages 260--265, 2016] proposed a polynomial time algorithm for maximizing the sum of radii (or perimeter) of non-overlapping balls or disks when the points are arbitrarily placed on a plane. We show that Eppstein's algorithm for maximizing sum of perimeter of the disks in { %DIFDELCMD < \IR%%% \mathbb{R }^2 gives a 2-approximation solution for the sum of area maximization problem. We propose a PTAS for our problem. These approximation results are extendible to higher dimensions . All these approximation results hold for the area maximization problem by regular convex polygons with even number of edges centered at the given points.
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we show that
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even for simplest objects like disks or squares
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convex