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The Maximum Distance-d Independent Set Problem on Unit Disk Graphs

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Book cover Frontiers in Algorithmics (FAW 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10823))

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Abstract

In this article, we study the maximum distance-d independent set problem, a variant of the maximum independent set problem, on unit disk graphs. We first show that the problem is NP-hard. Next, we propose a polynomial-time constant-factor approximation algorithm and a PTAS for the problem.

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Notes

  1. 1.

    The length of a smallest cycle.

  2. 2.

    By independent we mean for any \(p_i \in H_i \cap P\) and \(p_j \in H_j \cap P\), \(p_i\) and \(p_j\) are distance-d independent and also, \(OPT^i \cap OPT^j = \emptyset \).

  3. 3.

    If there are two components of \(G_\chi \) having distance less than d in G, then we can view them as a single component.

References

  1. Agnarsson, G., Damaschke, P., Halldórsson, M.M.: Powers of geometric intersection graphs and dispersion algorithms. Discrete Appl. Math. 132(1), 3–16 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM (JACM) 45(3), 501–555 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biedl, T., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang, G.J., Nemhauser, G.L.: The \(k\)-domination and \(k\)-stability problems on sun-free chordal graphs. SIAM J. Algebraic Discrete Methods 5(3), 332–345 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Das, G.K., De, M., Kolay, S., Nandy, S.C., Sur-Kolay, S.: Approximation algorithms for maximum independent set of a unit disk graph. Inf. Process. Lett. 115(3), 439–446 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34(6), 1302–1323 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eto, H., Guo, F., Miyano, E.: Distance-\(d\) independent set problems for bipartite and chordal graphs. J. Comb. Optim. 27(1), 88–99 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eto, H., Ito, T., Liu, Z., Miyano, E.: Approximability of the distance independent set problem on regular graphs and planar graphs. In: Chan, T.-H.H., Li, M., Wang, L. (eds.) COCOA 2016. LNCS, vol. 10043, pp. 270–284. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48749-6_20

    Chapter  Google Scholar 

  10. Eto, H., Ito, T., Liu, Z., Miyano, E.: Approximation algorithm for the distance-3 independent set problem on cubic graphs. In: Poon, S.-H., Rahman, M.S., Yen, H.-C. (eds.) WALCOM 2017. LNCS, vol. 10167, pp. 228–240. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53925-6_18

    Chapter  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness, vol. 29. W. H. Freeman, New York (2002)

    Google Scholar 

  13. Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete problems. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 47–63. ACM (1974)

    Google Scholar 

  14. Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1(2), 180–187 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Halldórsson, M.M.: Approximating discrete collections via local improvements. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 160–169. Society for Industrial and Applied Mathematics (1995)

    Google Scholar 

  16. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM (JACM) 32(1), 130–136 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hsu, W.L., Nemhauser, G.L.: Algorithms for minimum covering by cliques and maximum clique in claw-free perfect graphs. Discrete Math. 37(2–3), 181–191 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jallu, R.K., Das, G.K.: Improved algorithm for maximum independent set on unit disk graph. In: Govindarajan, S., Maheshwari, A. (eds.) CALDAM 2016. LNCS, vol. 9602, pp. 212–223. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29221-2_18

    Chapter  Google Scholar 

  19. Marathe, M.V., Breu, H., Hunt, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25(2), 59–68 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matsui, T.: Approximation algorithms for maximum independent set problems and fractional coloring problems on unit disk graphs. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 194–200. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-540-46515-7_16

    Chapter  Google Scholar 

  21. Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory Ser. B 28(3), 284–304 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Montealegre, P., Todinca, I.: On distance-d independent set and other problems in graphs with “few” minimal separators. In: Heggernes, P. (ed.) WG 2016. LNCS, vol. 9941, pp. 183–194. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53536-3_16

    Chapter  Google Scholar 

  23. Murphy, O.J.: Computing independent sets in graphs with large girth. Discrete Appl. Math. 35(2), 167–170 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nandy, S.C., Pandit, S., Roy, S.: Faster approximation for maximum independent set on unit disk graph. Inf. Process. Lett. 127, 58–61 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nieberg, T., Hurink, J., Kern, W.: A robust PTAS for maximum weight independent sets in unit disk graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 214–221. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30559-0_18

    Chapter  Google Scholar 

  26. Poljak, S.: A note on stable sets and colorings of graphs. Commentationes Math. Univ. Carol. 15(2), 307–309 (1974)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Indian Institute of Technology, Guwahati, India

    Sangram K. Jena & Gautam K. Das

  2. Indian Statistical Institute, Kolkata, India

    Ramesh K. Jallu & Subhas C. Nandy

Authors
  1. Sangram K. Jena
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  2. Ramesh K. Jallu
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  3. Gautam K. Das
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  4. Subhas C. Nandy
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Corresponding author

Correspondence to Gautam K. Das .

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Editors and Affiliations

  1. Texas A&M University, College Station, Texas, USA

    Jianer Chen

  2. Shanghai University of Finance and Economics, Shanghai, China

    Pinyan Lu

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Jena, S.K., Jallu, R.K., Das, G.K., Nandy, S.C. (2018). The Maximum Distance-d Independent Set Problem on Unit Disk Graphs. In: Chen, J., Lu, P. (eds) Frontiers in Algorithmics. FAW 2018. Lecture Notes in Computer Science(), vol 10823. Springer, Cham. https://doi.org/10.1007/978-3-319-78455-7_6

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  • DOI: https://doi.org/10.1007/978-3-319-78455-7_6

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