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

Published online by Cambridge University Press:  05 September 2011

Marcia R. Cerioli ,
Luerbio Faria ,
Talita O. Ferreira  and
Fábio Protti
Marcia R. Cerioli
Affiliation:
Instituto de Matemática and COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Brazil. Partially supported by CNPq and FAPERJ. cerioli@cos.ufrj.br
Luerbio Faria
Affiliation:
FFP, Universidade do Estado do Rio de Janeiro, Brazil. Partially supported by CNPq. luerbio@cos.ufrj.br
Talita O. Ferreira
Affiliation:
COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Brazil. Supported by CNPq. talita@cos.ufrj.br
Fábio Protti
Affiliation:
Instituto de Computação, Universidade Federal Fluminense, Brazil. Partially supported by CNPq and FAPERJ. fabio@ic.uff.br
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Abstract

A unit disk graph is the intersection graphof 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 setin 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 partitionin a penny graph is also NP-hard, and presenttwo linear-time approximation algorithms for the computation of clique partitions:a 3-approximation algorithm for unit disk graphsand a 2-approximation algorithm for penny graphs.

Keywords

Unit disk graphsunit coin graphspenny graphsindependent setclique partitionapproximation algorithms
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 45 , Issue 3 , July 2011 , pp. 331 - 346
Copyright
© EDP Sciences, 2011

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References

Baker, B.S., Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41 (1994) 153–180. CrossRef
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).
H. Breu, Algorithmic Aspects of Constrained Unit Disk Graphs. Ph.D. thesis, University of British Columbia (1996).
Breu, H. and Kirkpatrick, D.G., Unit disk graph recognition is NP-hard. Computational Geometry 9 (1998) 3–24. CrossRef
A. Borodin, I. Ivan, Y. Ye and B. Zimny, On sum coloring and sum multi-coloring for restricted families of graphs. Manuscript available at http://www.cs.toronto.edu/~bor/2420f10/stacs.pdf consulted 30 July 2011.
Clark, B.N., Colbourn, C.J. and Johnson, D.S., Unit disk graphs. Discrete Math. 86 (1990) 165–177. CrossRef
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) 73–79. CrossRef
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.
Erdös, P., On some problems of elementary and combinatorial geometry. Ann. Math. Pura Appl. Ser. 103 (1975) 99–108. CrossRef
Lichtenstein, D., Planar formulae and their uses. SIAM J. Comput. 43 (1982) 329–393. CrossRef
Garey, M.R. and Johnson, D.S., The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32 (1977) 826–834. CrossRef
M.R. Garey and D.S. Johnson, Computers and Intractability: a Guide to the Theory of NP-completeness. Freeman, New York (1979).
Golumbic, M.C., The complexity of comparability graph recognition and coloring. Computing 18 (1977) 199–208. CrossRef
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980).
Harborth, H., Lösung zu problem 664a. Elem. Math. 29 (1974) 14–15.
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) 238–274. CrossRef
Jansen, K. and Müller, H., The minimum broadcast time problem for several processor networks. Theor. Comput. Sci. 147 (1995) 69–85. CrossRef
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) 59–68. CrossRef
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) 194–200. CrossRef
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.
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) 311–322. CrossRef
Reutter, O., Problem 664a. Elem. Math. 27 (1972) 19.
J.P. Spinrad, Efficient Graph Representations, Fields Institute Monographs 19. American Mathematical Society (2003).
Valiant, L.G., Universality considerations in VLSI circuits. IEEE Trans. Comput. 30 (1981) 135–140. CrossRef