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Exact Algorithms for L(2,1)-Labeling of Graphs

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Abstract

The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O *((k+1)n) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k=4, where the running time of our algorithm is O(1.3006n). Furthermore we show that dynamic programming can be used to establish an O(3.8730n) algorithm to compute an optimal L(2,1)-labeling.

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References

  1. Alon, N., Wormald, N.: High degree graphs contain large-star factors (submitted)

  2. Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science 2006, pp. 575–582 (2006)

  3. Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J.: Approximations for lambda-colorings of graphs. Comput. J. 47, 193–204 (2004)

    Article  MATH  Google Scholar 

  4. Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM J. Discrete Math. 9, 309–316 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fiala, J., Kratochvíl, J.: Complexity of partial covers of graphs. In: Proceedings of ISAAC 2001. LNCS, vol. 2223, pp. 537–549. Springer, Berlin (2001)

    Google Scholar 

  6. Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discuss. Math. Graph Theory 22, 89–99 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of λ-labelings. Discrete Appl. Math. 113, 59–72 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fiala, J., Golovach, P., Kratochvíl, J.: Distance constrained labelings of graphs of bounded treewidth. In: Proceedings of ICALP 2005. LNCS, vol. 3580, pp. 360–372. Springer, Berlin (2005)

    Google Scholar 

  9. Fiala, J., Kratochvíl, J., Pór, A.: On the computational complexity of partial covers of theta graphs. Electron. Notes Discrete Math. 19, 79–85 (2005)

    Article  Google Scholar 

  10. Fomin, F., Grandoni, F., Kratsch, D.: Measure and conquer: Domination—a case study. In: Proceedings of ICALP 2005. LNCS, vol. 3380, pp. 192–203. Springer, Berlin (2005)

    Google Scholar 

  11. Fomin, F., Heggernes, P., Kratsch, D.: Exact algorithms for graph homomorphisms. Theory Comput. Syst. 41, 381–393 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gonçalves, D.: On the L(p,1)-labelling of graphs. In: Discrete Mathematics and Theoretical Computer Science Proceedings, vol. AE, pp. 81–86

  13. Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Discrete Math. 5, 586–595 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for L(2,1)-labeling of trees. arXiv:0810.0906v1 (2008)

  15. Havet, F., Reed, B., Sereni, J.-S.: L(2,1)-labellings of graphs. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms 2008, pp. 621–630 (2008)

  16. Hell, P., Nešetřil, J.: On the complexity of H-colouring. J. Comb. Theory Ser. B 48, 92–110 (1990)

    Article  MATH  Google Scholar 

  17. Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27, 119–123 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  18. Koivisto, M.: An O(2n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science 2006, pp. 583–590 (2006)

  19. Král’, D.: Channel assignment problem with variable weights. SIAM J. Discrete Math. 20, 690–704 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kratochvíl, J., Kratsch, D., Liedloff, M.: Exact algorithms for L(2,1)-labeling of graphs. In: Proceedings of MFCS 2007. LNCS, vol. 4708, pp. 513–524. Springer, Berlin (2007)

    Google Scholar 

  21. Leese, R.A., Noble, S.D.: Cyclic labellings with constraints at two distances. Electron. J. Comb. 11 (2004) (Research paper 16)

  22. Liu, D., Zhu, X.: Circular distance two labelings and circular chromatic numbers. Ars Comb. 69, 177–183 (2003)

    MATH  MathSciNet  Google Scholar 

  23. Liu, D., Zhu, X.: Multilevel distance labelings for paths and cycles. SIAM J. Discrete Math. 19, 610–621 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Roberts, F.S. Private communication to J. Griggs

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Authors and Affiliations

  1. Projet Mascotte I3S (CNRS & UNSA) and INRIA, INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902, Sophia-Antipolis Cedex, France

    Frédéric Havet

  2. Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Martin Klazar & Jan Kratochvíl

  3. Laboratoire d’Informatique Théorique et Appliquée, Université Paul Verlaine–Metz, 57045, Metz Cedex 01, France

    Dieter Kratsch

  4. Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, 45067, Orléans Cedex 2, France

    Mathieu Liedloff

Authors
  1. Frédéric Havet
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  2. Martin Klazar
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  3. Jan Kratochvíl
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  4. Dieter Kratsch
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  5. Mathieu Liedloff
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Corresponding author

Correspondence to Dieter Kratsch.

Additional information

The extended abstract of this paper was presented at the conference “Mathematical Foundations of Computer Science (MFCS 2007)”, Český Krumlov, Czech Republic, 26–31 August 2007 [20].

F. Havet was partially supported by the European project FET-AEOLUS.

M. Klazar and J. Kratochvíl was supported by Research grant 1M0545 of the Czech Ministry of Education.

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Havet, F., Klazar, M., Kratochvíl, J. et al. Exact Algorithms for L(2,1)-Labeling of Graphs. Algorithmica 59, 169–194 (2011). https://doi.org/10.1007/s00453-009-9302-7

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  • DOI: https://doi.org/10.1007/s00453-009-9302-7

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