Skip to main content
SpringerLink
Log in

Multicolor Discrepancy of Arithmetic Structures

  • Chapter
  • First Online:
A Panorama of Discrepancy Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2107))

  • 1535 Accesses

Abstract

In this chapter we present developments over the last 20 years in the discrepancy theory for hypergraphs with arithmetic structures, e.g. arithmetic progressions in the first N integers and their various generalizations, like Cartesian products, sums of arithmetic progressions, central arithmetic progressions in \(\mathbb{Z}_{p}\) and linear hyperplanes in finite vector spaces. We adopt the notion of multicolor discrepancy and show how the 2-color theory generalizes to multicolors exhibiting new phenomena at several places not visible in the 2-color theory, for example in the coloring of products of hypergraphs. The focus of the chapter is on proofs of lower bounds for the multicolor discrepancy for hypergraphs with arithmetic structures. Here, the application of Fourier analysis or linear algebra techniques is often not sufficient and has to be combined with combinatorial arguments, in the form of an interplay between the examination of suitable color classes and the Fourier analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The degree of a vertex is the number of hyperedges that contain the vertex.

  2. 2.

    A Hadamard matrix of dimension n is a matrix in { − 1, 1}n×n whose rows are mutually orthogonal.

References

  1. N. Alon, J.H. Spencer, P. Erdős, The Probabilistic Method (Wiley, New York, 1992)

    MATH  Google Scholar 

  2. K. Appel, W. Haken, Every planar map is four colorable. I: Discharging. Ill. J. Math. 21, 429–490 (1977)

    MathSciNet  MATH  Google Scholar 

  3. K. Appel, W. Haken, J. Koch, Every planar map is four colorable. II: Reducibility. Ill. J. Math. 21, 491–567 (1977)

    MathSciNet  MATH  Google Scholar 

  4. L. Babai, T.P. Hayes, P.G. Kimmel, The cost of the missing bit: Communication complexity with help, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, Dallas, Texas, USA, May 1998 (STOC 1998), 1998, pp. 673–682

    Google Scholar 

  5. W. Banaszczyk, Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Struct. Algorithm 12(4), 351–360 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bansal, Constructive algorithms for discrepancy minimization, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, Las Vegas, Nevada, USA, October 2010 (FOCS 2010), 2010, pp. 3–10

    Google Scholar 

  7. N. Bansal, J.H. Spencer, Deterministic discrepancy minimization, Demetrescu, C. (ed.) et al., Algorithms – ESA 2011, 19th annual European symposium, Saarbrücken, Germany, September 5–9, 2011, Proceedings. (Springer, Berlin). Lecture Notes in Computer Science 6942, 408–420, 2011. doi:10.1007/978-3-642-23719-5_35

  8. I. Bárány, B. Doerr, Balanced partitions of vector sequences. Lin. Algebra Appl. 414(2–3), 464–469 (2006). doi:10.1016/j.laa.2005.10.024

    Article  MATH  Google Scholar 

  9. I. Bárány, V.S. Grinberg, On some combinatorial questions in finite dimensional spaces. Lin. Algebra Appl. 41, 1–9 (1981)

    Article  MATH  Google Scholar 

  10. J. Beck, Roth’s estimate of the discrepancy of integer sequences is nearly sharp. Combinatorica 1, 319–325 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Beck, W.W.L. Chen, Irregularities of distribution. Cambridge Tracts in Mathematics, vol. 89 (Cambridge University Press, Cambridge, 1987)

    Google Scholar 

  12. J. Beck, T. Fiala, “Integer making” theorems. Discrete Appl. Math. 3(1), 1–8 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Beck, V.T. Sós, Discrepancy Theory, in Handbook of Combinatorics, ed. by R.L. Graham, M. Grötschel, L. Lovász (Elsevier, Amsterdam, 1995), pp. 1405–1446. Chap. 26

    Google Scholar 

  14. J. Beck, J.H. Spencer, Integral approximation sequences. Math. Program. 30, 88–98 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Beck, J.H. Spencer, Well distributed 2-colorings of integers relative to long arithmetic progressions. Acta Arithmetica 43, 287–298 (1984)

    MathSciNet  MATH  Google Scholar 

  16. T. Bohman, R. Holzman, Linear versus hereditary discrepancy. Combinatorica 25(1), 39–47 (2005). doi:10.1007/s00493-005-0003-9

    Article  MathSciNet  MATH  Google Scholar 

  17. S.A. Cook, The complexity of theorem-proving procedures, ACM, Proc. 3rd ann. ACM Sympos. Theory Computing, Shaker Heights, Ohio 1971, 1971, pp. 151–158

    Google Scholar 

  18. M. Charikar, A. Newman, A. Nikolov, Tight hardness results for minimizing discrepancy, in Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. SODA ’11 (SIAM, Philadelphia, 2011), pp. 1607–1614

    Google Scholar 

  19. B. Chazelle, The Discrepancy Method (Cambridge University Press, Cambridge, 2000)

    Book  MATH  Google Scholar 

  20. J. Cilleruelo, N. Hebbinghaus, Discrepancy in generalized arithmetic progressions. Eur. J. Combinator. 30(7), 1607–1611 (2009). doi:10.1016/j.ejc.2009.03.006

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Doerr, Linear discrepancy of totally unimodular matrices, in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, USA, January 2001 (SODA 2001), 2001, pp. 119–125

    Google Scholar 

  22. B. Doerr, Balanced coloring: Equally easy for all numbers of colors?, in Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science, Juan les Pins, France, March 2002 (STACS 2002), ed. by H. Alt, A. Ferreira, Lecture Notes in Computer Science, vol. 2285 (Springer, Berlin-Heidelberg, 2002), pp. 112–120

    Google Scholar 

  23. B. Doerr, Discrepancy in different numbers of colors. Discrete Math. 250, 63–70 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Doerr, Linear discrepancy of totally unimodular matrices. Combinatorica 24, 117–125 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Doerr, l p linear discrepancy of totally unimodular matrices. Lin. Algebra Appl. 420(2–3), 663–666 (2007). doi:10.1016/j.laa.2006.08.019

    Google Scholar 

  26. B. Doerr, A. Srivastav, Multi-color discrepancies. Combinator. Probab. Comput. 12, 365–399 (2003). doi:10.1017/S0963548303005662

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Doerr, M. Gnewuch, N. Hebbinghaus, Discrepancy of symmetric products of hypergraphs. Electron. J. Combinator. 13(1) (2006)

    Google Scholar 

  28. B. Doerr, N. Hebbinghaus, S. Werth, Improved bounds and schemes for the declustering problem. Theor. Comput. Sci. 359, 123–132 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Doerr, A. Srivastav, P. Wehr, Discrepancy of cartesian products of arithmetic progressions. Electron. J. Combinator. 11(1), 16 (2004). http://www.informatik.uni-kiel.de/~discopt/person/asr/publication/ap-electr04asrbedpetr.ps

  30. P. Erdős, Some unsolved problems. Mich. Math. J. 4(3), 291–300 (1957). doi:10.1307/mmj/1028997963

    Article  Google Scholar 

  31. P. Erdős, J.H. Spencer, Probabilistic Methods in Combinatorics (Akadémia Kiadó, Budapest, 1974)

    Google Scholar 

  32. G.A. Freiman, Elements of a structural theory of sumsets. (Kazan’: Kazan. Gosudarstv. Ped. Inst; Elabuzh. Gosudarstv. Ped. Inst., n.A., 1966)

    Google Scholar 

  33. A. Giannopoulos, On some vector balancing problems. Studia Math. 122(3), 225–234 (1997)

    MathSciNet  MATH  Google Scholar 

  34. E.D. Gluskin, Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Math. USSR Sbornik 64(1), 85–96 (1989). doi:10.1070/SM1989v064n01ABEH003295

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Gonthier, Formal proof - the four color theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)

    MathSciNet  MATH  Google Scholar 

  36. R.L. Graham, B.L. Rothschild, J.H. Spencer, Ramsey Theory (Wiley, New York, 1990)

    MATH  Google Scholar 

  37. D. Haussler, E. Welzl, \(\varepsilon\)-nets and simplex range queries. Discrete Comput. Geometry 2, 127–151 (1987)

    Google Scholar 

  38. N. Hebbinghaus, Discrepancy of arithmetic structures, PhD thesis, Christian-Albrechts-Universität Kiel, Technische Fakultät, 2005. http://eldiss.uni-kiel.de/macau/receive/dissertation_diss_00001851

  39. N. Hebbinghaus, Discrepancy of sums of two arithmetic progressions. Electron. Notes Discrete Math. 29, 547–551 (2007). doi:10.1016/j.endm.2007.07.087

    Article  Google Scholar 

  40. N. Hebbinghaus, A. Srivastav, Discrepancy of (centered) arithmetic progressions in \(\mathbb{Z}_{p}\). Eur. J. Combinator. 35, 324–334 (2014). doi:10.1016/j.ejc.2013.06.039

    Article  MathSciNet  MATH  Google Scholar 

  41. N. Hebbinghaus, T. Schoen, A. Srivastav, One-Sided Discrepancy of linear hyperplanes in finite vector spaces, in Analytic Number Theory, Essays in Honour of Klaus Roth, ed. by W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, R.C. Vaughan (Cambridge University Press, Cambridge, 2009), pp. 205–223. https://www.informatik.uni-kiel.de/~discopt/person/asr/publication/fq2rhneasrtos05.ps

  42. M. Helm, On the Beck-Fiala theorem. Discrete Math. 207(1–3), 73–87 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. R.M. Karp, Reducibility among combinatorial problems, (Russian. English original). Kibern. Sb., Nov. Ser. 12, 16-38 (1975); translation from Complexity of Computer Computations 1972, Plenum Press, New York, 1973, pp. 85–103

    Google Scholar 

  44. L. Kliemann, O. Kliemann, C. Patvardhan, V. Sauerland, A. Srivastav, A New QEA Computing Near-Optimal Low-Discrepancy Colorings in the Hypergraph of Arithmetic Progressions, in Proceedings of the 12th International Symposium on Experimental and Efficient Algorithms, Rome, Italy, June 2013 (SEA 2013), ed. by V. Bonifaci, C. Demetrescu, A. Marchetti-Spaccamela, Lecture Notes in Computer Science (Springer, Heidelberg, 2013), pp. 67–78. doi:10.1007/978-3-642-38527-8_8

  45. P. Knieper, Discrepancy of Arithmetic Progressions, PhD thesis, Institut für Informatik, Humboldt-Universität zu Berlin, 1997

    Google Scholar 

  46. R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, 2nd edn. (Cambridge University Press, Cambridge, 1994)

    Book  MATH  Google Scholar 

  47. L. Lovász, Coverings and colorings of hypergraphs, Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, 1973

    Google Scholar 

  48. L. Lovász, J.H. Spencer, K. Vesztergombi, Discrepancies of set-systems and matrices. Eur. J. Combinator. 7(2), 151–160 (1986)

    Article  MATH  Google Scholar 

  49. S. Lovett, R. Meka, Constructive discrepancy minimization by walking on the edges. CoRR abs/1203.5747 (2012)

    Google Scholar 

  50. J. Matoušek, Tight upper bound for the discrepancy of half-spaces. Discrete Comput. Geometry 13, 593–601 (1995)

    Article  MATH  Google Scholar 

  51. J. Matoušek, Geometric Discrepancy (Springer, Heidelberg, New York, 1999)

    Book  MATH  Google Scholar 

  52. J. Matoušek, On the linear and hereditary discrepancies. Eur. J. Combinator. 21(4), 519–521 (2000)

    Article  MATH  Google Scholar 

  53. J. Matoušek, Geometric Discrepancy (Springer, Heidelberg, New York, 2010)

    MATH  Google Scholar 

  54. J. Matoušek, The determinant bound for discrepancy is almost tight. Proc. Am. Math. Soc. 141(2), 451–460 (2013). doi:10.1090/S0002-9939-2012-11334-6

    Article  MATH  Google Scholar 

  55. J. Matoušek, A. Nikolov, Combinatorial Discrepancy for Boxes via the Ellipsoid-Infinity Norm. arXiv: 1408.1376 (2014)

    Google Scholar 

  56. J. Matoušek, J.H. Spencer, Discrepancy in arithmetic progressions. J. Am. Math. Soc. 9(1), 195–204 (1996). http://www.jstor.org/stable/2152845

  57. J. Matoušek, E. Welzl, L. Wernisch, Discrepancy and approximations for bounded VC-dimension. Combinatorica 13(4), 455–466 (1993). doi:10.1007/BF01303517

    Article  MathSciNet  MATH  Google Scholar 

  58. K. Mulmuley, Computational Geometry: An Introduction through Randomized Algorithms (Prentice-Hall, Englewood Cliffs, 1994)

    Google Scholar 

  59. M.B. Nathanson, Long arithmetic progressions and powers of 2, Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 735–739 (1989)

    Google Scholar 

  60. A. Nikolov, The Komlós conjecture holds for vector colorings. CoRR abs/1301.4039 (2013). http://arxiv.org/abs/1301.4039

  61. A. Nikolov, K. Talwar, L. Zhang, The geometry of differential privacy: the sparse and approximate cases, in Proceedings of the 45th ACM Symposium on Theory of Computing Conference, Palo Alto, CA, USA, June 2013 (STOC 2013), ed. by D. Boneh, T. Roughgarden, J. Feigenbaum, 2013, pp. 351–360

    Google Scholar 

  62. A. Přívětivý, Discrepancy of sums of three arithmetic progressions. Electron. J. Combinator. 13(1), 49 (2006)

    Google Scholar 

  63. A. Přívětivý, Coloring Problems in Geometric Context, PhD thesis, Charles University Prague, Faculty of Mathematics and Physics, 2007

    Google Scholar 

  64. R. Rado, Verallgemeinerung eines Satzes von van der Waerden mit Anwendungen auf ein Problem der Zahlentheorie. Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse 16/17, 589–596 (1933)

    Google Scholar 

  65. J. Riordan, An Introduction to Combinatorial Analysis (Wiley, New York, 1958)

    MATH  Google Scholar 

  66. N. Robertson, D.P. Sanders, P. Seymour, R. Thomas, The four-colour theorem. J. Combinator. Theory Ser. B 70(1), 2–44 (1997). doi:10.1006/jctb.1997.1750

    Article  MathSciNet  MATH  Google Scholar 

  67. K.F. Roth, Remark concerning integer sequences. Acta Arithmetica 9, 257–260 (1964)

    MathSciNet  MATH  Google Scholar 

  68. W. Rudin, Fourier Analysis on Groups (Wiley, New York, 1962)

    MATH  Google Scholar 

  69. V. Sauerland, Algorithm Engineering for some Complex Practice Problems, PhD thesis, Christian-Albrechts-Universität Kiel, Technische Fakultät, 2012. http://eldiss.uni-kiel.de/macau/receive/dissertation_diss_00009452

  70. J.H. Spencer, Six standard deviations suffice. Trans. Am. Math. Soc. 289(2), 679–706 (1985)

    Article  MATH  Google Scholar 

  71. J.H. Spencer, Ten Lectures on the Probabilistic Method, 2nd edn. (CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, 1994)

    Google Scholar 

  72. A. Srinivasan, Improving the discrepancy bound for sparse matrices: better approximations for sparse lattice approximation problems, in Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, USA, January 1997 (SODA 1997), 1997, pp. 692–701

    Google Scholar 

  73. A. Srivastav, Derandomization in Combinatorial Optimization, in Handbook of Randomized Computing, Vol. II, ed. by S. Rajasekaran, P.M. Pardalos, J.H. Reif, J.D.P. Rolim (Academic Publisher, Dordrecht, 2001), pp. 731–842. http://www.informatik.uni-kiel.de/~discopt/person/asr/publication/kluwerartikel.ps

  74. A. Srivastav, P. Stangier, Algorithmic Chernoff-Hoeffding inequalities in integer programming. Random Struct. Algorithm 1(8), 27–58 (1996). doi:10.1002/(SICI)1098-2418(199601)8:1¡27::AID-RSA2¿3.0.CO;2-T

    Article  MathSciNet  Google Scholar 

  75. B.L. van der Waerden, Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wiskunde 15, 212–216 (1927)

    Google Scholar 

Download references

Acknowledgements

We thank Mayank Singhal, MSc, for reading this chapter and Dr. Volkmar Sauerland for his assistance in Latex.

Author information

Authors and Affiliations

  1. Department of Computer Science, Kiel University, Kiel, Germany

    Nils Hebbinghaus & Anand Srivastav

Authors
  1. Nils Hebbinghaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anand Srivastav
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anand Srivastav .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Macquarie University, Sydney, New South Wales, Australia

    William Chen

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Anand Srivastav

  3. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milano, Italy

    Giancarlo Travaglini

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hebbinghaus, N., Srivastav, A. (2014). Multicolor Discrepancy of Arithmetic Structures. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_5

Download citation

Publish with us

Policies and ethics

Search

Navigation