Advertisement

Balanced k-Colorings

  • Therese C. Biedl
  • Eowyn Cenek
  • Timothy M. Chan
  • Erik D. Demaine
  • Martin L. Demaine
  • Rudolf Fleischer
  • Ming-Wei Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)

Abstract

While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k ≥ 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4d - 3, where d (the “dimension”) is the maximum number of subsets containing a common vertex. For 2-colorings, the bound on the discrepancy is at most max{2d-3, 2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete.

Keywords

Balance Equation Discrepancy Theory Rectangular Grid Grid Line Common Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley, New York, 1992. Chapter 12, pages 185–196.zbMATHGoogle Scholar
  2. 2.
    Jin Akiyama and Jorge Urrutia. A note on balanced colourings for lattice points. Discrete Mathematics, 83(1):123–126, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Tetsuo Asano, Tomomi Matsui, and Takeshi Tokuyama. On the complexities of the optimal rounding problems of sequences and matrices. In Proceedings of the 7th Scandinavian Workshop on Algorithm Theory (SWAT’00), Bergen, Norway, July 2000. To appear.Google Scholar
  4. 4.
    József Beck. Some results and problems in “combinatorial discrepancy theory”. In Topics in Classical Number Theory: Proceedings of the International Conference on Number Theory, pages 203–218, Budapest, Hungary, July 1981. Appeared in Colloquia Mathematica Societatis János Bolyai, volume 34, 1994.Google Scholar
  5. 5.
    József Beck and Tibor Fiala. “Integer-making” theorems. Discrete Applied Mathematics, 3:1–8, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    József Beck and Vera T. Sós. Discrepancy theory. In Handbook of Combinatorics, volume 2, pages 1405–1446. Elsevier, Amsterdam, 1995.Google Scholar
  7. 7.
    A.J.W. Hilton and D. de Werra. A sufficient condition for equitable edge-colourings of simple graphs. Discrete Mathematics, 128(1–3): 179–201, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 216–226, San Diego, California, May 1978.Google Scholar
  9. 9.
    Jiří Šíma. Table rounding problem. Comput. Artificial Intelligence, 18(3): 175–189, 1999.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Joel Spencer. Geometric discrepancy theory. Contemporary Mathematics, 223, 1999.Google Scholar
  11. 11.
    Joel Spencer. Personal communication. 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Therese C. Biedl
    • 1
  • Eowyn Cenek
    • 1
  • Timothy M. Chan
    • 1
  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Rudolf Fleischer
    • 1
  • Ming-Wei Wang
    • 1
  1. 1.Department of Computer ScienceUniversity of WaterlooUK

Personalised recommendations