A semidefinite programming hierarchy for packing problems in discrete geometry

Abstract

Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre’s semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Assume \(\theta _1 < \theta _2\) and let \(\epsilon \) be some number strictly between \((1 - \theta _1/\theta _2)/2\) and \(1\). Let \(D = (0, 1)\), and let \(d((x,i), (y,j))\) be given by \(\epsilon \delta _{i \ne j} + (1 - \epsilon \delta _{i \ne j}) \arccos (x \cdot y)\, (\theta _1+\theta _2)^{-1}\) when \(x \cdot y < \cos (\theta _i + \theta _j)\) and \(1\) otherwise.

  2. 2.

    Consider the graph with vertex set \([0, 1] \times \mathbb {Z}\) where \((x, i)\) and \((y, j)\) are adjacent if \(i = j\) or when \(x\) and \(y\) are both strictly smaller than \(|i-j|^{-1}\) (for \(i \ne j\)). Here each finite clique is contained in an open clique, but the countable clique \(\{0\} \times \mathbb {Z}\) is not.

  3. 3.

    In this paper cones are always assumed to be convex.

  4. 4.

    We show this in Remark 3.

References

  1. 1.

    Arens, R.: Topologies for homeomorphism groups. Am. J. Math. 68, 593–610 (1946)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Bachoc, C., Gijswijt, D.C., Schrijver, A., Vallentin, F.: Invariant semidefinite programs. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 219–269. Springer, Berlin (2012). http://arxiv.org/abs/1007.2905

  3. 3.

    Bachoc, C., Vallentin, F.: New upper bounds for kissing numbers from semidefinite programming. J. Am. Math. Soc. 21, 909–924 (2008). http://arxiv.org/abs/math/0608426

  4. 4.

    Bachoc, C., Nebe, G., de Oliveira Filho, F.M., Vallentin, F.: Lower bounds for measurable chromatic numbers. Geom. Funct. Anal. 19, 645–661 (2009). http://arxiv.org/abs/0801.1059

  5. 5.

    Barvinok, A.: A Course in Convexity. Grad. Stud. Math., vol. 54, American Mathematical Society (2002)

  6. 6.

    Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Springer, Berlin (1984)

    Google Scholar 

  7. 7.

    Borsuk, K., Ulam, S.: On symmetric products of topological spaces. Bull. Am. Math. Soc. 37, 875–882 (1931)

    Article  MathSciNet  Google Scholar 

  8. 8.

    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  9. 9.

    Cohn, H., Elkies, N.D.: New upper bounds on sphere packings I. Ann. Math. 157, 689–714 (2003). http://arxiv.org/abs/math/0110009

  10. 10.

    Cohn, H., Woo, J.: Three-point bounds for energy minimization. J. Am. Math. Soc. 25, 929–958 (2012). http://arxiv.org/abs/1103.0485

  11. 11.

    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. vi+97 (2012)

  12. 12.

    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geometriae Dedicata 6, 363–388 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Dieudonné, J.: Sur la séparation des ensembles convexes. Math. Ann. 163, 1–3 (1966)

    Article  MathSciNet  Google Scholar 

  14. 14.

    de Laat, D., de Oliveira Filho, F.M., Vallentin, F.: Upper Bounds for Packings of Spheres of Several Radii. Forum Math. (2012, to appear). http://arxiv.org/abs/1206.2608

  15. 15.

    de Oliveira Filho, F.M., Vallentin, F.: Computing Upper Bounds for Packing Densities of Congruent Copies of a Convex Body I. (preprint) (2013). http://arxiv.org/abs/1308.4893

  16. 16.

    Folland, G.B.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

    Google Scholar 

  17. 17.

    Gijswijt, D., Schrijver, A., Tanaka, H.: New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Comb. Theory Ser. A 113, 1719–1731 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Gijswijt, D., Mittelmann, H.D., Schrijver, A.: Semidefinite code bounds based on quadruple distances. IEEE Trans. Inf. Theory 58, 2697–2705 (2012). http://arxiv.org/abs/1005.4959

  19. 19.

    Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Handel, D.: Some homotopy properties of spaces of finite subsets of topological spaces. Houst. J. Math. 26, 747–764 (2000)

    MATH  MathSciNet  Google Scholar 

  21. 21.

    Hoffman, A.J.: On eigenvalues and colorings of graphs. In B. Harris (Ed.), Graph Theory and Its Applications, pp. 79–91. Academic Press, London (1970)

  22. 22.

    Kabatiansky, G.A., Levenshtein, V.I.: On bounds for packings on a sphere and in space. Probl. Peredachi Inf. 14, 3–25 (1978)

    Google Scholar 

  23. 23.

    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

  24. 24.

    Klee Jr, V.L.: Separation properties of convex cones. Proc. Am. Math. Soc. 6, 313–318 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optim. 12, 756–769 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28, 470–496 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    Laurent, M.: Strengthened semidefinite programming bounds for codes. Math. Program. Ser. B 109, 239–261 (2007)

  28. 28.

    Lindström, B.: Determinants on semilattices. Proc. Am. Math. Soc. 20, 207–208 (1969)

    Article  MATH  Google Scholar 

  29. 29.

    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Th. 25, 1–7 (1979)

    Article  MATH  Google Scholar 

  30. 30.

    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  31. 31.

    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Ind. Univ. Math. J. 42, 969–984 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Regts, G.: Upper Bounds for Ternary Constant Weight Codes from Semidefinite Programming and Representation Theory. Master thesis, University of Amsterdam (2009)

  33. 33.

    Schrijver, A.: A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inf. Theory 25, 425–429 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)

    Google Scholar 

  35. 35.

    Schrijver, A.: New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inf. Theory 51, 2859–2866 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. 36.

    Torquato, S., Jiao, Y.: Dense packings of the Platonic and Archimedean solids. Nature 460, 876–879 (2009)

    Article  Google Scholar 

  37. 37.

    Wilf, H.S.: Hadamard determinants, Möbius functions, and the chromatic number of a graph. Bull. Am. Math. Soc. 74, 960–964 (1968)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank Evan DeCorte and Cristóbal Guzmán for very helpful discussions. We also thank the referee whose suggestions helped to improve the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to David de Laat.

Additional information

The authors were supported by Vidi Grant 639.032.917 from the Netherlands Organization for Scientific Research (NWO).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

de Laat, D., Vallentin, F. A semidefinite programming hierarchy for packing problems in discrete geometry. Math. Program. 151, 529–553 (2015). https://doi.org/10.1007/s10107-014-0843-4

Download citation

Keywords

  • Lasserre hierarchy
  • Weighted independence number (stability number)
  • Infinite graphs
  • Geometric packing problems
  • Moment measures

Mathematics Subject Classification

  • 90C22
  • 52C17