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A 116/13-Approximation Algorithm for L(2, 1)-Labeling of Unit Disk Graphs

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SOFSEM 2019: Theory and Practice of Computer Science (SOFSEM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11376))

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Abstract

Given a graph, an L(2, 1)-labeling of the graph is an assignment \(\ell \) from the vertex set to the set of nonnegative integers such that for any pair of vertices (u, v), \(|\ell (u) - \ell (v)| \ge 2\) if u and v are adjacent, and \(\ell (u) \ne \ell (v)\) if u and v are at distance 2. The L(2, 1)-labeling problem is to minimize the span of \(\ell \) (i.e., \(\max _{u \in V}(\ell (u)) - \min _{u \in V}(\ell (u)) + 1\)). In this paper, we propose a new polynomial-time 116/13-approximation algorithm for L(2, 1)-labeling of unit disk graphs. This improves the previously best known ratio 12.

This work is partially supported by KAKENHI 17K19960, 17H01698.

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References

  1. Bodlaender, H.L., Kloks, T., Tan, R.B., Van Leeuwen, J.: Approximations for \(\lambda \)-colorings of graphs. Comput. J. 47(2), 193–204 (2004)

    Article  Google Scholar 

  2. Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. 9(1), 3–24 (1998)

    Article  MathSciNet  Google Scholar 

  3. Calamoneri, T.: The \({L} (h, k)\)-labelling problem: an updated survey and annotated bibliography. Comput. J. 54(8), 1344–1371 (2011)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1), 165–177 (1990)

    Article  MathSciNet  Google Scholar 

  6. Fiala, J., Fishkin, A.V., Fomin, F.: On distance constrained labeling of disk graphs. Theor. Comput. Sci. 326(1), 261–292 (2004)

    Article  MathSciNet  Google Scholar 

  7. Fiala, J., Golovach, P.A., Kratochvíl, J.: Distance constrained labelings of graphs of bounded treewidth. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 360–372. Springer, Heidelberg (2005). https://doi.org/10.1007/11523468_30

    Chapter  Google Scholar 

  8. Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of \(\lambda \)-labelings. Discrete Appl. Math. 113(1), 59–72 (2001)

    Article  MathSciNet  Google Scholar 

  9. Gonçalves, D.: On the \({L}(p,1)\)-labelling of graphs. Discrete Math. 308(8), 1405–1414 (2008). Third European Conference on Combinatorics

    Article  MathSciNet  Google Scholar 

  10. Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Discrete Math. 5(4), 586–595 (1992)

    Article  MathSciNet  Google Scholar 

  11. Hale, W.K.: Frequency assignment: theory and applications. Proc. IEEE 68(12), 1497–1514 (1980)

    Article  Google Scholar 

  12. Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for \({L} (2, 1)\)-labeling of trees. Algorithmica 66(3), 654–681 (2013)

    Article  MathSciNet  Google Scholar 

  13. Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: Algorithmic aspects of distance constrained labeling: a survey. Int. J. Netw. Comput. 4(2), 251–259 (2014)

    Article  Google Scholar 

  14. Junosza-Szaniawski, K., Rzażewski, P., Sokół, J., Wesek, K.: Coloring and \(L(2, 1)\)-labeling of unit disk intersection graphs. In: European Workshop on Computational Geometry (EuroCG), pp. 83–86 (2016)

    Google Scholar 

  15. Koller, A.E.: The frequency assignment problem. Ph.D. thesis. Brunel University, School of Information Systems, Computing and Mathematics (2005)

    Google Scholar 

  16. Roberts, F.S.: \({T}\)-colorings of graphs: recent results and open problems. Discrete Math. 93(2), 229–245 (1991)

    Article  MathSciNet  Google Scholar 

  17. Shao, Z., Yeh, R.K., Poon, K.K., Shiu, W.C.: The \({L}(2,1)\)-labeling of \({K}_{1, n}\)-free graphs and its applications. Appl. Math. Lett. 21(11), 1188–1193 (2008)

    Article  MathSciNet  Google Scholar 

  18. Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Math. 306(12), 1217–1231 (2006)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Informatics, Nagoya University, Nagoya, 464-8601, Japan

    Hirotaka Ono & Hisato Yamanaka

Authors
  1. Hirotaka Ono
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  2. Hisato Yamanaka
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Corresponding authors

Correspondence to Hirotaka Ono or Hisato Yamanaka .

Editor information

Editors and Affiliations

  1. University of Genoa, Genoa, Italy

    Barbara Catania

  2. Comenius University, Bratislava, Slovakia

    Rastislav Královič

  3. Poznań University of Technology, Poznań, Poland

    Jerzy Nawrocki

  4. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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Ono, H., Yamanaka, H. (2019). A 116/13-Approximation Algorithm for L(2, 1)-Labeling of Unit Disk Graphs. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_30

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  • DOI: https://doi.org/10.1007/978-3-030-10801-4_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10800-7

  • Online ISBN: 978-3-030-10801-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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