Cutting cycles of rods in space: hardness and approximation

We study the problem of cutting a set of rods (line segments in ℝ3) into fragments, using a minimum number of cuts, so that the resulting set of fragments admits a depth order. We prove that this problem is NP-complete, even when the rods have only three distinct orientations. We also give a polynomial-time approximation algorithm with no restriction on rod orientation that computes a solution of size O(τ log τ log log τ), where τ is the size of an optimal solution.

[1]  Joseph Naor,et al.  Approximating Minimum Feedback Sets and Multicuts in Directed Graphs , 1998, Algorithmica.

[2]  Micha Sharir,et al.  Efficient Hidden Surface Removal for Objects with Small Union Size , 1992, Comput. Geom..

[3]  M. Carter.  Computer graphics: Principles and practice , 1997 .

[4]  Alexandra Solan,et al.  Cutting cylces of rods in space , 1998, SCG '98.

[5]  Mark de Berg,et al.  Computing and verifying depth orders , 1992, SCG '92.

[6]  San Skulrattanakulchai,et al.  4-edge-coloring Graphs of Maximum Degree 3 in Linear Time , 2002, Inf. Process. Lett..

[7]  Leonidas J. Guibas,et al.  Counting and cutting cycles of lines and rods in space , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[8]  Micha Sharir,et al.  Online Point Location in Planar Arrangements and Its Applications , 2001, SODA '01.

[9]  Matthew J. Kaltz.  3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects , 1997 .

[10]  Micha Sharir,et al.  Cutting Triangular Cycles of Lines in Space , 2005, Discret. Comput. Geom..

[11]  Matthew J. Katz.  3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects , 1997, Comput. Geom..

[12]  Mark de Berg,et al.  Ray Shooting, Depth Orders and Hidden Surface Removal , 1993, Lecture Notes in Computer Science.

[13]  Roberto Tamassia,et al.  On Embedding a Graph in the Grid with the Minimum Number of Bends , 1987, SIAM J. Comput..

[14]  James C. Miller,et al.  Computer graphics principles and practice, second edition , 1992, Comput. Graph..

[15]  Joseph Naor,et al.  Divide-and-conquer approximation algorithms via spreading metrics , 2000, JACM.

[16]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

[17]  Mark de Berg,et al.  Efficient ray shooting and hidden surface removal , 1991, SCG '91.

[18]  Mark de Berg,et al.  Vertical ray shooting and computing depth orders of fat objects , 2008 .

[19]  Micha Sharir,et al.  A simple output-sensitive algorithm for hidden surface removal , 1992, TOGS.