Integer programming approaches for minimum stabbing problems
RAIRO - Operations Research - Recherche Opérationnelle, Volume 48 (2014) no. 2, pp. 211-233.

The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associated to set of points in the plane. The complexity of the mstr remains open whilst the other two are known to be 𝓝𝓟-hard. This paper presents integer programming (ip) formulations for these three problems, that allowed us to solve them to optimality through ip branch-and-bound (b&b) or branch-and-cut (b&c) algorithms. Moreover, these models are the basis for the development of Lagrangian heuristics. Computational tests were conducted with instances taken from the literature where the performance of the Lagrangian heuristics were compared with that of the exact b&b and b&c algorithms. The results reveal that the Lagrangian heuristics yield solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times. To our knowledge, this is the first computational study ever reported in which these three stabbing problems are considered and where provably optimal solutions are given.

DOI: 10.1051/ro/2014008
Classification: 90C10, 90C27, 90C47, 90C57, 90C59
Keywords: integer programming, lagrangian relaxation, stabbing problems, branch-and-bound, branch-and-cut
@article{RO_2014__48_2_211_0,
     author = {Piva, Breno and de Souza, Cid C. and Frota, Yuri and Simonetti, Luidi},
     title = {Integer programming approaches for minimum stabbing problems},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {211--233},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {2},
     year = {2014},
     doi = {10.1051/ro/2014008},
     mrnumber = {3264376},
     zbl = {1295.90021},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro/2014008/}
}
TY  - JOUR
AU  - Piva, Breno
AU  - de Souza, Cid C.
AU  - Frota, Yuri
AU  - Simonetti, Luidi
TI  - Integer programming approaches for minimum stabbing problems
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2014
SP  - 211
EP  - 233
VL  - 48
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2014008/
DO  - 10.1051/ro/2014008
LA  - en
ID  - RO_2014__48_2_211_0
ER  - 
%0 Journal Article
%A Piva, Breno
%A de Souza, Cid C.
%A Frota, Yuri
%A Simonetti, Luidi
%T Integer programming approaches for minimum stabbing problems
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2014
%P 211-233
%V 48
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2014008/
%R 10.1051/ro/2014008
%G en
%F RO_2014__48_2_211_0
Piva, Breno; de Souza, Cid C.; Frota, Yuri; Simonetti, Luidi. Integer programming approaches for minimum stabbing problems. RAIRO - Operations Research - Recherche Opérationnelle, Volume 48 (2014) no. 2, pp. 211-233. doi : 10.1051/ro/2014008. http://archive.numdam.org/articles/10.1051/ro/2014008/

[1] P. Agarwal, B. Aronov and S. Suri, Stabbing triangulations by lines in 3D, in Proceedings of the eleventh annual symposium on Computational geometry, SCG '95, New York, NY, USA. ACM (1995) 267-276.

[2] J. Beasley, Lagrangean relaxation, in Modern Heuristic Techniques for Combinatorial Problems. McGraw-Hill (1993) 243-303.

[3] R. Beirouti and J. Snoeyink, Implementations of the LMT heuristic for minimum weight triangulation, in Proceedings of the Fourteenth Annual Symposium on Computational Geometry, SCG '98, New York, NY, USA ACM (1998) 96-105.

[4] M. Berg and M. Kreveld, Rectilinear decompositions with low stabbing number. Infor. Proc. Lett. 52 (1994) 215-221. | MR | Zbl

[5] J.A. De Loera, S. Hosten, F. Santos and B. Sturmfels, The polytope of all triangulations of a point configuration. Documenta Math. 1 (1996) 103-119. | MR | Zbl

[6] E. Demaine, J. Mitchell and J. O'Rourke, The open problems project. Available online (acessed in January 2010). http://maven.smith.edu/˜orourke/TOPP/.

[7] M.T. Dickerson and M.H. Montague, A (usually) connected subgraph of the minimum weight triangulation, in Proceedings of the 12th Annual ACM Symposyum on Computational Geometry (1996) 204-213.

[8] J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices. J. Res. Nat. Bur. Stand. B 69 (1965) 125-130. | MR | Zbl

[9] S. Fekete, M. Lübbecke and H. Meijer, Minimizing the stabbing number of matchings, trees, and triangulations, in SODA edited by J. Munro. SIAM (2004) 437-446. | MR | Zbl

[10] S. Fekete, M. Lübbecke and H. Meijer, Minimizing the stabbing number of matchings, trees, and triangulations. Discrete Comput. Geometry 40 (2008) 595-621. | MR | Zbl

[11] M. Fischetti, J.J.S. Gonzalez and P. Toth, Solving the orienteering problem through branch-and-cut. INFORMS J. Comput. 10 133-148, 1998. | MR | Zbl

[12] M. Grötschel and O. Holland, Solving matching problems with linear programming. Math. Program. 33 (1985) 243-259. | MR | Zbl

[13] T. Koch and A. Martin, Solving Steiner tree problems in graphs to optimality. Networks 33 (1998) 207-232. | MR | Zbl

[14] V. Kolmogorov, Blossom V: a new implementation of a minimum cost perfect matching algorithm. Math. Program. Comput. 1 (2009) 43-67. | MR | Zbl

[15] T.L. Magnanti and L.A. Wolsey, Optimal trees. Handbooks in Operations Research and Management Science 7 (1995) 503-615. | MR | Zbl

[16] J. Mitchell and J. O'Rourke, Computational geometry. SIGACT News 32 (2001) 63-72.

[17] J. Mitchell and E. Packer, Computing geometric structures of low stabbing number in the plane, in Proc. 17th Annual Fall Workshop on Comput. Geometry and Visualization. IBM Watson (2007).

[18] W. Mulzer, Index of /m˜ulzer/pubs/mwtsoftware/old/ipelets. Available online (accessed in March 2011). http://page.mi.fu-berlin.de/mulzer/pubs/mwt˙software/old/ipelets/LMTSkeleton.tar.gz.

[19] W. Mulzer and G. Rote, Minimum-weight triangulation is NP-hard. J. ACM 55 (2008) 1-11. | MR

[20] A.P. Nunes, Uma abordagem de programação inteira para o problema da triangulação de custo mínimo. Master's thesis, Institute of Computing, University of Campinas, Campinas, Brazil (1997). In Portuguese.

[21] M.W. Padberg and M.R. Rao, Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7 (1982) 67-80. | MR | Zbl

[22] B. Piva and C.C. De Souza, The minimum stabbing triangulation problem: IP models and computational evaluation. ISCO (2012) 36-47. | MR

[23] G. Reinelt, TSPLIB. Available online (acessed in March 2011). http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/.

[24] J.R. Shewchuk, Stabbing Delaunay tetrahedralizations. Discrete and Comput. Geometry 32 (2002) 343. | MR | Zbl

[25] M. Solomon, VRPTW benchmark problems. Available online (acessed in August 2011). http://w.cba.neu.edu/˜msolomon/problems.htm.

[26] C.D. Tóth, Orthogonal subdivisions with low stabbing numbers, Vol. 3608. Lect. Notes in Comput. Sci. Springer, Berlin/Heidelberg (2005) 256-268. | MR | Zbl

[27] L.A. Wolsey, Integer Programming. John Wiley & Sons (1998). | MR | Zbl

Cited by Sources: