Abstract
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. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of vertices. The complexity of finding a spanning tree of minimum stabbing number is one of the original 30 questions on “The Open Problems Project” list of outstanding problems in computational geometry by Demaine, Mitchell, and O’Rourke.
We show \(\mathcal{N}\mathcal{P}\) -hardness of stabbing problems by means of a general proof technique. For matchings, this also implies a nontrivial lower bound on the approximability. On the positive side, we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. From the corresponding linear programming relaxation we obtain polynomial-time lower bounds and show that there always is an optimal fractional solution that contains an edge of at least constant weight. We conjecture that the resulting iterated rounding scheme constitutes a constant-factor approximation algorithm.
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Agarwal, P.K.: Ray shooting and other applications of spanning trees with low stabbing number. SIAM J. Comput. 21(3), 540–570 (1992)
Agarwal, P.K., Aronov, B., Suri, S.: Stabbing triangulations by lines in 3D. In: Proc. 11th ACM Sympos. Computational Geometry, pp. 267–276 (1995)
Arkin, E.M., Bender, M.A., Demaine, E., Fekete, S.P., Mitchell, J.S.B., Sethia, S.: Optimal covering tours with turn costs. SIAM J. Comput. 35, 531–566 (2005)
Aronov, B., Fortune, S.: Approximating minimum-weight triangulations in three dimensions. Discrete Comput. Geom. 21(4), 527–549 (1999)
Aronov, B., Brönnimann, H., Chang, A.Y., Chiang, Y.-J.: Cost-driven octree construction schemes: an experimental study. Comput. Geom. Theory Appl. 31, 127–148 (2005)
Chazelle, B., Welzl, E.: Quasi-optimal range searching in space of finite vc-dimension. DISCG 4, 467–489 (1989)
de Berg, M., van Kreveld, M.: Rectilinear decompositions with low stabbing number. Inf. Process. Lett. 52(4), 215–221 (1994)
Demaine, E.D., Mitchell, J.S.B., O’Rourke, J.: The open problems project. http://cs.smith.edu/~orourke/TOPP/Welcome.html (2003)
Edmonds, J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Natl. Burean Stand. 69B, 125–130 (1965)
Fekete, S.P.: On simple polygonalizations with optimal area. Discrete Comput. Geom. 23, 73–110 (2000)
Fekete, S.P., Lübbecke, M.E., Meijer, H.: Minimizing the stabbing number of matchings, spanning trees, and triangulations. In Proc. 15th ACM-SIAM Sympos. Discrete Algorithms, pp. 430–439 (2004)
Fekete, S.P., Lübbecke, M.E., Meijer, H.: Computing structures of minimum stabbing number. Technical report (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability—A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)
Held, M., Klosowski, J.T., Mitchell, J.S.B.: Evaluation of collision detection methods for virtual reality fly-throughs. In: Proc. 7th Canadian Conf. Computational Geometry, pp. 205–210 (1995)
Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: shoot a ray, take a walk. J. Algorithms 18, 403–431 (1995)
Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)
Jain, K.: Personal communication (2003)
Magnanti, T.L., Wolsey, L.A.: Optimal trees. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds.) Network Models. Handbooks in Operations Research and Management Science, vol. 7, pp. 503–616. North-Holland, Amsterdam (1995)
Matoušek, J.: Spanning trees with low crossing number. Inf. Theor. Appl. 25, 102–123 (1991)
Mitchell, J.S.B., O’Rourke, J.: Computational geometry column 42. Int. J. Comput. Geom. Appl. 11(5), 573–582 (2001)
Padberg, M., Rao, M.R.: Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7, 67–80 (1982)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
Shewchuk, J.R.: Stabbing Delaunay tetrahedralizations. Discrete Comput. Geom. 32(3), 339–343 (2004)
Tóth, C.: Orthogonal subdivisions with low stabbing numbers. In: Proc. 9th International Workshop on Algorithms and Data Structures (WADS 2005). LNCS, vol. 3608, pp. 256–268. Springer, Berlin (2005)
Welzl, E.: On spanning trees with low crossing numbers. In: Monien, B., Ottmann, T. (eds.) Data Structures and Efficient Algorithms. LNCS, vol. 594. Springer, Berlin (1992)
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An extended abstract appeared in the Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms [11].
M.E. Lübbecke visits to Kingston and Stony Brook were supported by a DFG travel grant.
H. Meijer partially supported by NSERC while visiting Braunschweig in 2002.
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Fekete, S.P., Lübbecke, M.E. & Meijer, H. Minimizing the Stabbing Number of Matchings, Trees, and Triangulations. Discrete Comput Geom 40, 595–621 (2008). https://doi.org/10.1007/s00454-008-9114-6
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DOI: https://doi.org/10.1007/s00454-008-9114-6