Abstract
The minimum stabbing triangulation of a set of points in the plane (mstr) was previously investigated in the literature. The complexity of the mstr remains open and, to our knowledge, no exact algorithm was proposed and no computational results were reported earlier in the literature of the problem. This paper presents integer programming (ip) formulations for the mstr, that allow us to solve it exactly through ip branch-and-bound (b&b) algorithms. Moreover, one of these models is the basis for the development of a sophisticated Lagrangian heuristic for the problem. Computational tests were conducted with two instance classes comparing the performance of the latter algorithm against that of a standard (exact) b&b. The results reveal that the Lagrangian algorithm yielded solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P., Aronov, B., Suri, S.: Stabbing triangulations by lines in 3d. In: Proceedings of the Eleventh Annual Symposium on Computational Geometry, SCG 1995, pp. 267–276. ACM, New York (1995)
Beasley, J.: Lagrangean relaxation. In: Modern Heuristic Techniques for Combinatorial Problems, pp. 243–303. McGraw-Hill (1993)
Beirouti, R., Snoeyink, J.: Implementations of the LMT heuristic for minimum weight triangulation. In: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, SCG 1998, pp. 96–105. ACM, New York (1998)
Berg, M., Kreveld, M.: Rectilinear decompositions with low stabbing number. Information Processing Letters 52(4), 215–221 (1994)
Demaine, E., Mitchell, J., O’Rourke, J.: The open problems project, http://maven.smith.edu/~orourke/TOPP/ (acessed in January 2010)
Dickerson, M.T., Montague, M.H.: A (usually) connected subgraph of the minimum weight triangulation. In: Proc. of the 12th Annual ACM Symp. on Comp. Geom, pp. 204–213 (1996)
Fekete, S., Lübbecke, M., Meijer, H.: Minimizing the stabbing number of matchings, trees, and triangulations. In: Munro, J. (ed.) SODA, pp. 437–446. SIAM (2004)
Fekete, S., Lübbecke, M., Meijer, H.: Minimizing the stabbing number of matchings, trees, and triangulations. Disc. Comp. Geometry 40, 595–621 (2008)
Mitchell, J., O’Rourke, J.: Computational geometry. SIGACT News 32(3), 63–72 (2001)
Mitchell, J., Packer, E.: Computing geometric structures of low stabbing number in the plane. In: Proc. 17th Annual Fall Work. on Comp. Geom. and Vis. IBM Watson (2007)
Mulzer, W.: LMT-skeleton program, http://page.mi.fu-berlin.de/mulzer/pubs/mwt_software/old/ipelets/LMTSk//eleton.tar.gz (accessed in March 2011)
Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. Journal of the ACM 55, 11:1–11:29 (2008)
Nunes, A.P.: Uma abordagem de programação inteira para o problema da triangulação de custo mÃnimo. Master’s thesis, Institute of Computing, University of Campinas, Campinas (1997) (in Portuguese)
Reinelt, G.: TSPLIB, http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/ (acessed in March 2011)
Shewchuk, J.R.: Stabbing Delaunay tetrahedralizations. Disc. Comp. Geometry 32, 343 (2002)
Tóth, C.D.: Orthogonal Subdivisions with Low Stabbing Numbers. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 256–268. Springer, Heidelberg (2005)
Wolsey, L.A.: Integer Programming. John Wiley and Sons (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Piva, B., de Souza, C.C. (2012). The Minimum Stabbing Triangulation Problem: IP Models and Computational Evaluation. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-32147-4_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32146-7
Online ISBN: 978-3-642-32147-4
eBook Packages: Computer ScienceComputer Science (R0)