Cutting Cycles of Rods in Space: Hardness and Approximation

We study the problem of cutting a set of rods (line segments in R-3) 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(tau log tau log log tau), where tau is the size of an optimal solution.