Abstract
This chapter surveys on-line and approximation algorithms for the maximum independent set and coloring problems on intersection graphs of disks. It includes a more detailed treatment of recent upper and lower bounds on the competitive ratio of on-line algorithms for coloring such graphs.
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References
Ambühl, C., Wagner, U.: On the clique problem in intersection graphs of ellipses. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 489–500. Springer, Heidelberg (2002)
Asplund, E., Grünbaum, B.: On a coloring problem. Mathematica Scandinavica 8, 181–188 (1960)
Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41(1), 153–180 (1994)
Berman, P., DasGupta, B., Muthukrishnan, S., Ramaswami, S.: Improved approximation algorithms for rectangle tiling and packing. In: Proceedings of the 12th Annual ACM–SIAM Symposium on Discrete Algorithms SODA 2001, pp. 427–436 (2001)
Breu, H., Kirkpatrick, D.G.: On the complexity of recognizing intersection and touching graphs of disks. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 88–98. Springer, Heidelberg (1996)
Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Computational Geometry: Theory and Applications 9(1–2), 3–24 (1998)
Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms 46, 178–189 (2003)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86(1–3), 165–177 (1990)
Erlebach, T., Fiala, J.: On-line coloring of geometric intersection graphs. Computational Geometry: Theory and Applications 23(2), 243–255 (2002)
Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric graphs. In: Proceedings of the 12th Annual ACM–SIAM Symposium on Discrete Algorithms SODA 2001, pp. 671–679 (2001)
Fiala, J., Fishkin, A.V., Fomin, F.V.: Off-line and on-line distance constrained labeling of graphs. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 464–475. Springer, Heidelberg (2001)
Gräf, A.: Coloring and recognizing special graph classes. Musikinformatik und Medientechnik Bericht 20/95, Johannes Gutenberg-Universität Mainz (1995)
Gräf, A., Stumpf, M., Weißenfels, G.: On coloring unit disk graphs. Algorithmica 20(3), 277–293 (1998); See also Musikinformatik und Medientechnik Bericht 17/94, Johannes Gutenberg-Universität Mainz (1994)
Gyárfás, A., Lehel, J.: On-line and first fit colourings of graphs. Jornal of Graph Theory 12(2), 217–227 (1988)
Hale, W.K.: Frequency assignment: Theory and applications. Proc. of the IEEE 68(12), 1497–1514 (1980)
Halldórsson, M.M.: Approximating discrete collections via local improvements. In: Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 160–169 (1995)
Hliněný, P., Kratochvíl, J.: Representing graphs by disks and balls. Discrete Mathematics 229(1–3), 101–124 (2001)
Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)
Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM 32(1), 130–136 (1985)
Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-Approximation schemes for NP- and PSPACE-hard problems for geometric graphs. Journal of Algorithms 26(2), 238–274 (1998)
Irani, S.: Coloring inductive graphs on-line. Algorithmica 11, 53–72 (1994)
Jansen, K.: Approximate strong separation with application in fractional graph coloring and preemptive scheduling. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 100–111. Springer, Heidelberg (2002)
Jansen, K., Porkolab, L.: On preemptive resource constrained scheduling: Polynomial-time approximation schemes. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 329–349. Springer, Heidelberg (2002)
Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akad. Leipzig 88, 141–164 (1936)
Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Applied Mathematics 52, 233–252 (1994)
Malesińska, E.: Graph theoretical models for frequency assignment problems. PhD thesis, Technical University of Berlin (1997)
Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)
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)
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)
Peeters, R.: On coloring j-unit sphere graphs. Technical report, Dept. of Economics, Tilburg University (1991)
Raghavan, V., Spinrad, J.: Robust algorithms for restricted domains. In: Proceedings of the 12th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 460–467 (2001)
Sachs, H.: Coin graphs, polyhedra, and conformal mapping. Discrete Mathematics 134, 133–138 (1994)
Wang, D., Kuo, Y.-S.: A study on two geometric location problems. Information Processing Letters 28, 281–286 (1988)
Wolff, A., Strijk, T.: The map-labeling bibliography, http://i11www.ira.uka.de/map-labeling/bibliography/
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Erlebach, T., Fiala, J. (2006). Independence and Coloring Problems on Intersection Graphs of Disks. In: Bampis, E., Jansen, K., Kenyon, C. (eds) Efficient Approximation and Online Algorithms. Lecture Notes in Computer Science, vol 3484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671541_5
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DOI: https://doi.org/10.1007/11671541_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32212-2
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