Approximating 3D points with cylindrical segments

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. Our PTAS is built upon a simple (though slower) approximation algorithm for the case when k = 1.