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On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs

Abstract

Rigid graph theory is an active area with many open problems, especially regarding embeddings in \({\mathbb R}^d\) or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system’s complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in \(\mathbb {C}^d\) and \(S^d\). The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension \(d\ge 5\), both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in \(S^2\) and \(\mathbb {C}^3\). We exploit Bernstein’s second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

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Notes

  1. APX is the class of all NP optimization problems, for which there exist polynomial-time approximation algorithms. This approximation is bounded by a constant (for further details see [1]).

  2. The graphs in this example and the following ones are named after their class (L for Laman and G for Geiringer) and the number of their embeddings in the correspondent euclidean space as in [3].

  3. The idea of using the product of polytopes is derived by a proof for the mixed volumes corresponding to the weighted m-Bézout bound in [38]

References

  1. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation, Combinatorial Optimization Problems and their Approximability Properties. Springer, Berlin (1999)

    MATH  Google Scholar 

  2. Baglivo, J., Graver, J.: Incidence and Symmetry in Design and Architecture. Cambridge Urban and Architectural Studies (No. 7). Cambridge University Press, Cambridge (1983)

    Google Scholar 

  3. Bartzos, E., Emiris, I.Z., Legerský, J., Tsigaridas, E.: On the maximal number of real embeddings of minimally rigid graphs in \(\mathbb{R}^2\), \(\mathbb{R}^3\) and \(S^2\). J. Symb. Comput. (2019). https://doi.org/10.1016/j.jsc.2019.10.015

    Article  MATH  Google Scholar 

  4. Bartzos, E., Legerský, J.: Graph embeddings in the plane, space and sphere–source code and results. Zenodo (2018). https://doi.org/10.5281/zenodo.1495153

    Article  Google Scholar 

  5. Bartzos, E., Schicho, J.: Source code and examples for the paper “On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs” (2019). https://doi.org/10.5281/zenodo.3542061

  6. Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Its Appl. 9(3), 183–185 (1975). https://doi.org/10.1007/BF01075595

    MathSciNet  Article  MATH  Google Scholar 

  7. Bihan, F., Sottile, F.: New fewnomial upper bounds from Gale dual polynomial system. Mosc. Math. J. 7(3), 387–407 (2007)

    MathSciNet  Article  Google Scholar 

  8. Billinge, S., Duxbury, P., Gonçalves, D., Lavor, C., Mucherino, A.: Assigned and unassigned distance geometry: applications to biological molecules and nanostructures. 4OR 14(4), 337–376 (2016)

    MathSciNet  Article  Google Scholar 

  9. Blumenthal, L.M.: Theory and Applications of Distance Geometry. Chelsea Publishing Company, New York (1970)

    MATH  Google Scholar 

  10. Borcea, C., Streinu, I.: On the number of embeddings of minimally rigid graphs. In: Proceedings of ACM Symposium on Computational Geometry, SCG ’02. ACM, pp. 25–32 (2002). https://doi.org/10.1145/513400.513404

  11. Brègman, L.: Some properties of nonnegative matrices and their permanents. Dokl. Akad. Nauk SSSR 211(1), 27–30 (1973)

    MathSciNet  MATH  Google Scholar 

  12. Busé, L., Emiris, I., Mourrain, B., Ruatta, O., Trébuchet, P.: Multires, a Maple package for multivariate resolution problems (2002). http://www-sop.inria.fr/galaad/logiciels/multires

  13. Capco, J., Gallet, M., Grasegger, G., Koutschan, C., Lubbes, N., Schicho, J.: The number of realizations of a Laman graph. SIAM J. Appl. Algebra Geom. 2(1), 94–125 (2018). https://doi.org/10.1137/17M1118312

    MathSciNet  Article  MATH  Google Scholar 

  14. Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Dickenstein, A., Emiris, I.: Multihomogeneous resultant formulae by means of complexes. J. Symb. Comput. 36(3), 317–342 (2003). https://doi.org/10.1016/S0747-7171(03)00086-5

    MathSciNet  Article  MATH  Google Scholar 

  16. Duff, T., Hill, C., Jensen, A., Lee, K., Leykin, A., Sommars, J.: Solving polynomial systems via homotopy continuation and monodromy. IMA J. Numer. Anal. 39(3), 1421–1446 (2018). https://doi.org/10.1093/imanum/dry017

    MathSciNet  Article  Google Scholar 

  17. Emiris, I., Mantzaflaris, A.: Multihomogeneous resultant formulae for systems with scaled support. J. Symb. Comput. 47(7), 820–842 (2012). https://doi.org/10.1016/j.jsc.2011.12.010. Special issue on: International Symposium on Symbolic and Algebraic Computation (ISSAC 2009)

    MathSciNet  Article  MATH  Google Scholar 

  18. Emiris, I., Tsigaridas, E., Varvitsiotis, A.: Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.) Distance Geometry: Theory, Methods and Applications, pp. 23–45. Springer, Berlin (2013)

    Chapter  Google Scholar 

  19. Emiris, I.Z., Vidunas, R.: Root counts of semi-mixed systems, and an application to counting nash equilibria. In: Proceedings of ACM International Symposium on Symbolic and Algebraic Computation, ISSAC, pp. 154–161. ACM (2014). https://doi.org/10.1145/2608628.2608679

  20. Emmerich, D.: Structures Tendues et Autotendantes. Ecole d’Architecture de Paris, la Villette (1988)

  21. Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Comb. 15(01), Research paper R77 (2008). https://doi.org/10.37236/801

  22. Gallet, M., Grasegger, G., Schicho, J.: Counting realizations of Laman graphs on the sphere. Electron. J. Comb. (2020). https://doi.org/10.37236/8548

    Article  MATH  Google Scholar 

  23. Gáspár, M., Csermely, P.: Rigidity and flexibility of biological networks. Brief. Funct. Genom. 11(6), 443–456 (2012). https://doi.org/10.1093/bfgp/els023

    Article  Google Scholar 

  24. Grasegger, G., Koutschan, C., Tsigaridas, E.: Lower bounds on the number of realizations of rigid graphs. Exp. Math. 29(2), 125–136 (2020). https://doi.org/10.1080/10586458.2018.1437851

    MathSciNet  Article  MATH  Google Scholar 

  25. Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity, Graduate Studies in Mathematics, vol. 2. American Mathematical Society, Providence (1993)

    MATH  Google Scholar 

  26. Harris, J., Tu, L.: On symmetric and skew-symmetric determinantal varieties. Topology 23, 71–84 (1984)

    MathSciNet  Article  Google Scholar 

  27. Jackson, B., Owen, J.: Equivalent realisations of a rigid graph. Discrete Appl. Math. 256, 42–58 (2019). https://doi.org/10.1016/j.dam.2017.12.009. Spec. Issue on Distance Geometry: Theory & Applications (DGTA 16)

    MathSciNet  Article  MATH  Google Scholar 

  28. Khovanskii, A.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Its Appl. 12(1), 38–46 (1978). https://doi.org/10.1007/BF01077562

    MathSciNet  Article  Google Scholar 

  29. Khovanskii, A.: Fewnomials, Translations of Mathematical Monographs, vol. 88. American Mathematical Society, Providence (1991)

    Google Scholar 

  30. Kouchnirenko, A.: Polydres de Newton et nombres de Milnor. Invent. Math. 32, 1–32 (1976)

    MathSciNet  Article  Google Scholar 

  31. Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4(4), 331–340 (1970)

    MathSciNet  Article  Google Scholar 

  32. Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Math. 308(8), 1425–1437 (2008). Spec. Issue on 3rd European Conf. Combinatorics

    MathSciNet  Article  Google Scholar 

  33. van Lint, J., Wilson, R.: A Course in Combinatorics. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  34. Maehara, H.: On Graver’s conjecture concerning the rigidity problem of graphs. Discrete Comput. Geometry 6, 339–342 (1991)

    MathSciNet  Article  Google Scholar 

  35. Malajovich, G., Meer, K.: Computing minimal multi-homogeneous bezout numbers is hard. Theory Comput. Syst. 40(4), 553–570 (2007). https://doi.org/10.1007/s00224-006-1322-y

    MathSciNet  Article  MATH  Google Scholar 

  36. Maxwell, J.: On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27, 294–299 (1864)

    Article  Google Scholar 

  37. Minc, H.: Upper bounds for permanents of \(\left(0,1 \right)\)-matrices. Bull. Am. Math. Soc. 69, 789–791 (1963). https://doi.org/10.1090/S0002-9904-1963-11031-9

    MathSciNet  Article  MATH  Google Scholar 

  38. Mondal, P.: How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity. arXiv:1806.05346v2 [math.AG] (2019)

  39. Nixon, A., Owen, J., Power, S.: A characterization of generically rigid frameworks on surfaces of revolution. SIAM J. Discrete Math. 28, 2008–2028 (2014). https://doi.org/10.1137/130913195

    MathSciNet  Article  MATH  Google Scholar 

  40. Pollaczek-Geiringer, H.: Über die Gliederung ebener Fachwerke. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 7(1), 58–72 (1927)

    Article  Google Scholar 

  41. Pollaczek-Geiringer, H.: Zur Gliederungstheorie räumlicher Fachwerke. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 12(6), 369–376 (1932)

    Article  Google Scholar 

  42. Robbins, H.: A remark on Stirling’s formula. Am. Math. Mon. 62(1), 26–29 (1955)

    MathSciNet  MATH  Google Scholar 

  43. Schrijver, A.: Bounds on the number of Eulerian orientations. Combinatorica 3(3), 375–380 (1983). https://doi.org/10.1007/BF02579193

    MathSciNet  Article  MATH  Google Scholar 

  44. Schulze, B., Whiteley, W.: Rigidity and Scene Analysis, chap. 61, pp. 1593–1632. CRC Press LLC, Boca Raton (2017)

    Google Scholar 

  45. Shafarevich, I.: Intersection Numbers, pp. 233–283. Springer, Berlin (2013)

    Google Scholar 

  46. Sommese, A., Wampler, C.I.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific Publishing, Singapore (2005)

    Book  Google Scholar 

  47. Steffens, R., Theobald, T.: Mixed volume techniques for embeddings of Laman graphs. Comput. Geom. 43, 84–93 (2010)

    MathSciNet  Article  Google Scholar 

  48. Tay, T.S., Whiteley, W.: Generating isostatic frameworks. Topologie Structurale pp. 21–69 (1985)

  49. Verschelde, J.: Modernizing PHCpack through phcpy. In: Proceedings of European Conference on Python in Science (EuroSciPy 2013), pp. 71–76 (2014)

  50. Walter, D., Husty, M.L.: On a 9-bar linkage, its possible configurations and conditions for paradoxical mobility. IFToMM World Congress, Besanon, France (2007)

    Google Scholar 

  51. Whiteley, W.: Cones, infinity and one-story buildings. Topol. Struct. 8, 53–70 (1983)

    MATH  Google Scholar 

  52. Whiteley, W.: Infinitesimally rigid polyhedra. I. Statics of frameworks. Trans. Am. Math. Soc. 285(2), 431–465 (1984). https://doi.org/10.2307/1999446

    MathSciNet  Article  MATH  Google Scholar 

  53. Zelazo, D., Franchi, A., Allgöwer, F., Bülthoff, H.H., Giordano, P.: Rigidity Maintenance Control for Multi-Robot Systems. Robotics: Science and Systems. Sydney, Australia (2012)

  54. Zhu, Z., So, A.C., Ye, Y.: Universal rigidity and edge sparsification for sensor network localization. SIAM J. Optim. 20(6), 3059–3081 (2010)

    MathSciNet  Article  Google Scholar 

  55. Ziegler, G.: Lectures on Polytopes. Graduate Texts in Mathematics. Springer, Berlin (1995)

    Book  Google Scholar 

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Acknowledgements

IZE and JS are partially supported, and EB is fully supported by project ARCADES which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675789. IZE and EB are members of team AROMATH, joint between INRIA Sophia-Antipolis and NKUA. We thank Georg Grasegger for providing us with runtimes of the combinatorial algorithm that calculates \(c_d(G)\) and for explaining to EB the counting of embeddings for triangle-free Geiringer graphs. We thank Jan Verschelde for help with applying our method to the Desargues graph, and Timothy Duff for giving us an example on the use of MonodromySolver to find the embeddings of the Icosahedron.

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Bartzos, E., Emiris, I.Z. & Schicho, J. On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs. AAECC 31, 325–357 (2020). https://doi.org/10.1007/s00200-020-00447-7

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  • DOI: https://doi.org/10.1007/s00200-020-00447-7

Keywords

  • Rigid graph
  • Laman graph
  • Multihomogeneous Bézout bound
  • Combinatorial algorithm
  • Permanent
  • Bernstein’s second theorem
  • Mixed volume

Mathematics Subject Classification

  • 52C25
  • 14N10
  • 52A39