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Mathematical Programming

, Volume 166, Issue 1–2, pp 273–295 | Cite as

Realizability and inscribability for simplicial polytopes via nonlinear optimization

  • Moritz Firsching
Full Length Paper Series A
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Abstract

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. In order to show non-realizability of simplicial spheres, we extend the method of finding biquadratic final polynomials for matroid polytopes to partial matroid polytopes. Combining these two methods we obtain a complete classification of neighborly polytopes of dimension 4, 6 and 7 with 11 vertices, of neighborly 5-polytopes with 10 vertices, as well as a complete classification of simplicial 3-spheres with 10 vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.

Mathematics Subject Classification

52B11 52C40 52B40 90C30 

Notes

Acknowledgements

I am indebted to Arnau Padrol, who started the project by telling me about the problem discussed in Sect. 3.4 and proposing to use optimization methods in order to solve it and to Hao Chen, who got very interested in proving non-inscribability results. I am thankful to Hiroyuki Miyata for the help while working on Sect. 3.1; Hiroyuki Miyata was kind enough to run the biquadratic final polynomial method for us until Arnau Padrol and I had implemented it and to run double checks afterwards. For the help in Sect. 3.2 I thank Frank Lutz, who provided the list of triangulations and David Bremner, who decided the existence of a compatible matroid polytope for these triangulations. I enjoyed discussions on the topic discussed in Sect. 3.3 with Florian Frick, John Sullivan and Frank Lutz, who again provided a list of triangulations. I would like to thank Moritz W. Schmitt, Philip Brinkmann and Louis Theran for valuable discussions and Benjamin Müller as well as Ambros Gleixner for help with my questions regarding SCIP. I thank two anonymous referees for their constructive suggestions for improvements. Last but not least, I am very grateful to Günter M. Ziegler for helping at every step in the preparation of this paper.

Funding was provided by Deutsche Forschungsgemeinschaft (Grant No. SFB/TRR109).

References

  1. 1.
    Achterberg, T., Berthold, T., Koch, T., Wolter, K.: Constraint integer programming: a new approach to integrate CP and MIP. In: Perron, L., Trick, M.A. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. Lecture Notes in Computer Science, vol. 5015, pp. 6–20. Springer, Berlin (2008)Google Scholar
  2. 2.
    Altshuler, A., Bokowski, J., Steinberg, L.: The classification of simplicial \(3\)-spheres with nine vertices into polytopes and nonpolytopes. Discrete Math. 31(2), 115–124 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Altshuler, A.: Neighborly 4-polytopes and neighborly combinatorial 3-manifolds with ten vertices. Can. J. Math. 29(225), 420 (1977)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Altshuler, A., McMullen, P.: The number of simplicial neighbourly \(d\)-polytopes with \(d + 3\) vertices. Mathematika 20(02), 263–266 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Adiprasito, K.A., Padrol, A., Theran, L.: Universality theorems for inscribed polytopes and Delaunay triangulations. Discrete Comput. Geom. 54(2), 412–431 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Altshuler, A., Steinberg, L.: Neighborly 4-polytopes with 9 vertices. J. Comb. Theory Ser A 15(3), 270–287 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Altshuler, A., Steinberg, L.: The complete enumeration of the 4-polytopes and 3-spheres with eight vertices. Pac. J. Math. 117(1), 1–16 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Baier, P.: NP-completeness of partial chirotope extendibility (2005). http://arxiv.org/abs/math/0504430
  10. 10.
    Bokowski, J., Bremner, D., Gévay, G.: Symmetric matroid polytopes and their generation. Eur. J. Comb. 30(8), 1758–1777 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bagchi, B., Datta, B.: A structure theorem for pseudomanifolds. Discrete Math. 188(1), 41–60 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bremner, D., Deza, A., Hua, W., Schewe, L.: More bounds on the diameters of convex polytopes. Optim. Methods Softw. 28(3), 442–450 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bokowski, J., Ewald, G., Kleinschmidt, P.: On combinatorial and affine automorphisms of polytopes. Israel J. Math. 47(2–3), 123–130 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bokowski, J., de Oliveira, A.G.: Simplicial convex \(4\)-polytopes do not have the isotopy property. Port. Math. 47(3), 309–318 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bagnara, R., Hill, P.M., Zaffanella, E.: The Parma Polyhedra Library: toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Sci. Comput. Program. 72(1–2), 3–21 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  17. 17.
    Bokowski, J., Richter, J.: On the finding of final polynomials. Eur. J. Comb. 11(1), 21–34 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Brown, K.Q.: Voronoi diagrams from convex hulls. Inf. Process. Lett. 9(5), 223–228 (1979)CrossRefzbMATHGoogle Scholar
  19. 19.
    Brückner, M.: Vielecke und Vielflache: Theorie und Geschichte. Teubner, Leipzig (1900)zbMATHGoogle Scholar
  20. 20.
    Brückner, M.: Über die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope), Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Afdeeling Natuurkunde. Eerste sectie, vol. 10, pp. 2–27. Johannes Müller, Amsterdam (1909)Google Scholar
  21. 21.
    Bokowski, J., Shemer, I.: Neighborly 6-polytopes with 10 vertices. Israel J. Math. 58(1), 103–124 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bokowski, J., Sturmfels, B.: Computational Synthetic Geometry. Lecture Notes in Mathematics, vol. 1355. Springer, Berlin (1989)Google Scholar
  23. 23.
    Bokowski, J., Schuchert, P.: Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes. Discrete Comput. Geom. 13(1), 347–361 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bremner, D., Schewe, L.: Edge-graph diameter bounds for convex polytopes with few facets. Exp. Math. 20(3), 229–237 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bender, E.A., Wormald, N.C.: The number of rooted convex polyhedra. Can. Math. Bull. 31(1), 99–102 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Dillencourt, M.B., Smith, W.D.: Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Math. 161(1–3), 63–77 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Donoho, D.L., Tanner, J.: Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102(27), 9452–9457 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Donoho, D.L., Tanner, J.: Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA 102(27), 9446–9451 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Eberhard, V.: Zur Morphologie der Polyeder. Teubner, Leipzig (1891)zbMATHGoogle Scholar
  30. 30.
    Euclid. Elementa Geometriae. Erhardus Ratdolt, Venice (1482)Google Scholar
  31. 31.
    Finschi, L.: A graph theoretical approach for reconstruction and generation of oriented matroids. Ph.D. thesis, Swiss Federal Institute of Technology Zurich (2001)Google Scholar
  32. 32.
    Finbow, W.: Simplicial neighbourly 5-polytopes with nine vertices. Bol. Soc. Mat. Mex. 21(1), 39–51 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Firsching, M.: Computing maximal copies of polyhedra contained in a polyhedron. Exp. Math. 24(1), 98–105 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Frick, F., Lutz, F.H, Sullivan, J.M.: Simplicial manifolds with small valence (in preparation)Google Scholar
  35. 35.
    Fukuda, K., Miyata, H., Moriyama, S.: Complete enumeration of small realizable oriented matroids. Discrete Comput. Geom. 49(2), 359–381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Frick, F.: A minimal irreducible triangulation of \(S^3\). In: Benedetti, B., Delucchi, E., Moci, L. (eds.) Proceedings of Combinatorial Methods in Topology and Algebra (CoMeTA), Cortona. Springer, Berlin (2013) (to appear) (extended abstract, 4 pages)Google Scholar
  37. 37.
    Frick, F: Combinatorial restrictions on cell complexes. Dissertation, TU Berlin (2015)Google Scholar
  38. 38.
    Fusy, É.: Counting \(d\)-polytopes with \(d+ 3\) vertices. Electron. J. Comb. 13(1), R23 (2006)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Gale, D.: Neighborly and cyclic polytopes. In: Klee, V. (eds.) Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, vol. 7, pp. 225–232. American Mathematical Society (1963)Google Scholar
  40. 40.
    Gonska, B., Padrol, A.: Neighborly inscribed polytopes and Delaunay triangulations (2015). arXiv:1308.5798v2
  41. 41.
    Grünbaum, B.: Convex Polytopes. Wiley, New York (1967)zbMATHGoogle Scholar
  42. 42.
    Grünbaum, B., Sreedharan, V.P.: An enumeration of simplicial 4-polytopes with 8 vertices. J. Comb. Theory 2(4), 437–465 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Gonska, B., Ziegler, G.M.: Inscribable stacked polytopes. Adv. Geom. 13(4), 723–740 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Hodgson, C.D., Rivin, I., Smith, W.D.: A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bull. Am. Math. Soc. 27(2), 246–251 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kalai, G.: Many triangulated spheres. Discrete Comput Geom. 3(1), 1–14 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kleinschmidt, P.: Sphären mit wenigen Ecken. Geom. Dedicata 5(3), 307–320 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Keith Lloyd, E.: The number of \(d\)-polytopes with \(d + 3\) vertices. Mathematika 17(01), 120–132 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
  49. 49.
    Lutz, F.H.: Combinatorial 3-manifolds with 10 vertices. Beitr. Algebra Geom. 49(1), 97–106 (2008)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Mani, P.: Spheres with few vertices. J. Comb. Theory Ser. A 13(3), 346–352 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Montellano-Ballesteros, J.J., Strausz, R.: Counting polytopes via the radon complex. J. Comb. Theory Ser. A 106(1), 109–121 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17(02), 179–184 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    McMullen, P.: The number of neighbourly \(d\)-polytopes with \(d + 3\) vertices. Mathematika 21(01), 26–31 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Miyata, H.: Database of neighborly polytopes. https://sites.google.com/site/hmiyata1984/neighborly_polytopes
  55. 55.
    Mnëv, N.E: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. Lecture Notes in Mathematics, vol. 1346, pp. 527–543. Springer, Berlin (1988)Google Scholar
  56. 56.
    Motzkin, T.S.: Comonotone curves and polyhedra. Bull. Am. Math. Soc. 63, 35 (1957)CrossRefGoogle Scholar
  57. 57.
    Miyata, H., Padrol, A.: Enumerating neighborly polytopes and oriented matroids. Exp. Math. 24(4), 489–505 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Padrol, A.: Many neighborly polytopes and oriented matroids. Discrete Comput. Geom. 50(4), 865–902 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Richter-Gebert, J.: Two interesting oriented matroids. Doc. Math. 1(137), 137–148 (1996)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Richter-Gebert, J., Ziegler, G.M.: Realization spaces of \(4\)-polytopes are universal. Bull. Am. Math. Soc. 32(4), 403–412 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Richter-Gebert, J., Ziegler, G.M.: Oriented matroids. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry (Chapter 6). CRC Press, Boca Raton (2004)Google Scholar
  62. 62.
    Richmond, L.B., Wormald, N.C.: The asymptotic number of convex polyhedra. Trans. Am. Math. Soc. 273 721–735 (1982)Google Scholar
  63. 63.
    William A.S. et al. Sage Mathematics Software (Version 6.2). The Sage Development Team (2014). http://www.sagemath.org
  64. 64.
    Schlegel, V.: Über Projectionsmodelle der regelmässigen vier-dimensionalen Körper. Waren (1886)Google Scholar
  65. 65.
    Schläfli, L.: In: Graf, J.H. (eds.) Theorie der vielfachen Kontinuität, number 38 in Denkschriften der Schweizerischen naturforschenden Gesellschaft. Zürcher & Furrer (1901)Google Scholar
  66. 66.
    Schmutz, E.: Rational points on the unit sphere. Open Math. 6(3), 482–487 (2008)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Schewe, L.: Nonrealizable minimal vertex triangulations of surfaces: showing nonrealizability using oriented matroids and satisfiability solvers. Discrete Comput. Geom. 43(2), 289–302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Shemer, I.: Neighborly polytopes. Israel J. Math. 43(4), 291–314 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Shor, P.: Stretchability of pseudolines is NP-hard. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 531–554. American Mathematical Society (1991)Google Scholar
  70. 70.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://oeis.org/
  71. 71.
    Steiner, J.: Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander, mit Berücksichtigung der Arbeiten alter und neuer Geometer über Porismen, Projections-Methoden, Geometrie der Lage, Transversalen, Dualität und Reciprocität, etc., vol. 1. Gustav Fincke (1832)Google Scholar
  72. 72.
    Steinitz, E.: Polyeder und Raumeinteilungen. In: Meyer, F., Mohrmann, H. (eds.) Encyclopädie der Mathematischen Wissenschaften. Geometrie, erster Teil, zweite Hälfte, vol. 3, pp. 1–139. Teubner, Leipzig (1922)Google Scholar
  73. 73.
    Sturmfels, B.: Boundary complexes of convex polytopes cannot be characterized locally. J. Lond. Math. Soc. 2(2), 314–326 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Sturmfels, B.: Neighborly polytopes and oriented matroids. Eur. J. Comb. 9(6), 537–546 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Tschirschnitz, F.: Testing extendability for partial chirotopes is NP-Complete. In: Proceedings of the 13th Canadian Conference on Computational Geometry, pp 165–168 (2001)Google Scholar
  76. 76.
    Tutte, W.T.: On the enumeration of convex polyhedra. J. Comb. Theory Ser. B 28(2), 105–126 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Vigerske, S., Gleixner, A.: SCIP: Global Optimization of Mixed-Integer Nonlinear Programs in a Branch-and-Cut Framework. Technical Report 16-24, ZIB, Berlin (2016)Google Scholar
  78. 78.
    Wächter, A., Biegler, T.L.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Wiener, C.: Über Vielecke und Vielflache. Teubner, Leipzig (1864)Google Scholar
  80. 80.
    Ziegler, G.M.: Lectures on Polytopes. Number 152 in Graduate Texts in Mathematics. Springer, Berlin (1995) (updated seventh printing 2007)Google Scholar

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Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany

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