Abstract
In this paper, we study a 3D geometric problem originated from computing neural maps in the computational biology community: Given a set S of n points in 3D, compute k cylindrical segments (with different radii, orientations, lengths and no segment penetrates another) enclosing S such that the sum of their radii is minimized. There is no known result in this direction except when k=1. The general problem is strongly NP-hard and we obtain a polynomial time approximation scheme (PTAS) for any fixed k>1 in O(n3k-2/δ4k-3) time by returning k cylindrical segments with sum of radii at most (1+δ) of the corresponding optimal value, if exist. Our PTAS is built upon a simple (though slower) approximation algorithm for the case when k=1.
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