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A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation

  • Marcia R. Cerioli (a1), Luerbio Faria (a2), Talita O. Ferreira (a3) and Fábio Protti (a4)


A unit disk graph is the intersection graph of a family of unit disks in the plane. If the disks do not overlap, it is also a unit coin graph or penny graph. It is known that finding a maximum independent set in a unit disk graph is a NP-hard problem. In this work we extend this result to penny graphs. Furthermore, we prove that finding a minimum clique partition in a penny graph is also NP-hard, and present two linear-time approximation algorithms for the computation of clique partitions: a 3-approximation algorithm for unit disk graphs and a 2-approximation algorithm for penny graphs.



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[1] Baker, B.S., Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41 (1994) 153180.
[2] P. Berman, M. Karpinski and A.D. Scott, Approximation hardness and satisfiability of bounded occurrence instances of SAT, in Electronic Colloquium on Computational Complexity – ECCC (2003).
[3] H. Breu, Algorithmic Aspects of Constrained Unit Disk Graphs. Ph.D. thesis, University of British Columbia (1996).
[4] Breu, H. and Kirkpatrick, D.G., Unit disk graph recognition is NP-hard. Computational Geometry 9 (1998) 324.
[5] A. Borodin, I. Ivan, Y. Ye and B. Zimny, On sum coloring and sum multi-coloring for restricted families of graphs. Manuscript available at consulted 30 July 2011.
[6] Clark, B.N., Colbourn, C.J. and Johnson, D.S., Unit disk graphs. Discrete Math. 86 (1990) 165177.
[7] Cerioli, M.R., Faria, L., Ferreira, T.O. and Protti, F., On minimum clique partition and maximum independent set in unit disk graphs and penny graphs: complexity and approximation. LACGA'2004 – Latin-American Conference on Combinatorics, Graphs and Applications. Santiago, Chile (2004). Electron. Notes Discrete Math. 18 (2004) 7379.
[8] T. Erlebach, K. Jansen and E. Seidel, Polynomial-time approximation schemes for geometric graphs, in Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (2000) 671–679.
[9] Erdös, P., On some problems of elementary and combinatorial geometry. Ann. Math. Pura Appl. Ser. 103 (1975) 99108.
[10] Lichtenstein, D., Planar formulae and their uses. SIAM J. Comput. 43 (1982) 329393.
[11] Garey, M.R. and Johnson, D.S., The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32 (1977) 826834.
[12] M.R. Garey and D.S. Johnson, Computers and Intractability: a Guide to the Theory of NP-completeness. Freeman, New York (1979).
[13] Golumbic, M.C., The complexity of comparability graph recognition and coloring. Computing 18 (1977) 199208.
[14] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980).
[15] Harborth, H., Lösung zu problem 664a. Elem. Math. 29 (1974) 1415.
[16] Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J. and Stearns, R.E., NC-Approximation schemes for NP- and PSPACE-Hard problems for geometric graphs. J. Algor. 26 (1998) 238274.
[17] Jansen, K. and Müller, H., The minimum broadcast time problem for several processor networks. Theor. Comput. Sci. 147 (1995) 6985.
[18] Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S. and Rosenkrantz, D.J., Simple heuristics for unit disk Graphs. Networks 25 (1995) 5968.
[19] Matsui, T., Approximation algorithms for maximum independent set problems and fractional coloring problems on unit disk graphs. In Discrete and Computational Geometry. Lect. Notes Comput. Sci. 1763 (2000) 194200.
[20] I.A. Pirwani and M.R. Salavatipour, A weakly robust PTAS for minimum clique partition in unit disk graphs (Extended Abstract), Proceedings of SWAT 2010. Lect. Notes Comput. Sci. 6139 (2010) 188–199.
[21] Pemmaraju, S.V. and Pirwani, I.A., Good quality virtual realization of unit ball graphs, Proceedings of the 15th Annual European Symposium on Algorithms. Lect. Notes Comput. Sci. 4698 (2007) 311322.
[22] Reutter, O., Problem 664a. Elem. Math. 27 (1972) 19.
[23] J.P. Spinrad, Efficient Graph Representations, Fields Institute Monographs 19. American Mathematical Society (2003).
[24] Valiant, L.G., Universality considerations in VLSI circuits. IEEE Trans. Comput. 30 (1981) 135140.


A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation

  • Marcia R. Cerioli (a1), Luerbio Faria (a2), Talita O. Ferreira (a3) and Fábio Protti (a4)


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