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Given $n$ non-vertical lines in 3-space, their vertical depth (above/below)relation can contain cycles. We show that the lines can be cut into$O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relationamong these pieces is now a proper partial order. This bound is nearly tight inthe worst case.  Previous results on this topic could only handle restricted cases of theproblem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir(2005), or only cycles in grid-like patterns, by Chazelle et al. (1992)), andthe bounds were considerably weaker---much closer to the trivial quadraticbound.  Our proof uses a recent variant of the polynomial partitioning technique, dueto Guth, and some simple tools from algebraic geometry. It is much morestraightforward than the previous "purely combinatorial" methods.  Our technique can be extended to eliminating all cycles in the depth relationamong segments, and among constant-degree algebraic arcs. We hope that asuitable extension of this technique could be used to handle the much moredifficult case of pairwise-disjoint triangles.  We also discuss several algorithms for constructing a small set of cuts so asto eliminate all depth-relation cycles among the lines (minimizing such a set,for the case of line segments, is known to be NP-complete). The performance ofthese algorithms improves due to our new bound, but, so far, none of them both(a) produce close to $n^{3/2}$ cuts, and (b) run in time close to $n^{3/2}$, inthe worst case.  Our results almost completely settle a 35-year-old open problem incomputational geometry, motivated by hidden-surface removal in computergraphics.

Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions