Ray shooting and other applications of spanning trees with low stabbing number
SIAM Journal on Computing
Rectilinear decompositions with low stabbing number
Information Processing Letters
A pedestrian approach to ray shooting: shoot a ray, take a walk
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
Stabbing triangulations by lines in 3D
Proceedings of the eleventh annual symposium on Computational geometry
Combinatorial optimization
Optimal covering tours with turn costs
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Computational geometry column 41
ACM SIGACT News
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On Spanning Trees with Low Crossing Numbers
Data Structures and Efficient Algorithms, Final Report on the DFG Special Joint Initiative
Computational Geometry: Theory and Applications
The asymmetric traveling salesman problem on graphs with bounded genus
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Orthogonal subdivisions with low stabbing numbers
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
The minimum stabbing triangulation problem: IP models and computational evaluation
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
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The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. We investigate problems of finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open problem; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke.We show that minimum stabbing problems are NP-complete. We also show that an iterated rounding technique is applicable for matchings and spanning trees of minimum stabbing number by showing that there is a polynomially solvable LP-relaxation that has fractional solutions with at least one heavy edge. This suggests constant-factor approximations. Our approach uses polyhedral methods that are related to another open problem (from a combinatorial optimization list), in combination with geometric properties. We also demonstrate that the resulting techniques are practical for actually solving problems with up to several hundred points optimally or near-optimally.