Abstract
We investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs (\(G_{n, m, \vec{p}}\)) which includes the model of [10] (the “uniform” random intersection graphs model) as an important special case. We also define an interesting variation of the model of random intersection graphs, similar in spirit to random regular graphs. (b) For this model we derive exact formulae for the mean and variance of the number of independent sets of size k (for any k) in the graph. (c) We then propose and analyse three algorithms for the efficient construction of large independent sets in this model. The first two are variations of the greedy technique while the third is a totally new algorithm. Our algorithms are analysed for the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters.
This work has been partially supported by the IST Programme of the European Union under contract numbers IST-2001-33116 (FLAGS) and 001907 (DELIS).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. John Wiley & Sons, Inc., Chichester (2000)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)
Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)
Díaz, J., Penrose, M.D., Petit, J., Serna, M.: Approximating Layout Problems on Random Geometric Graphs. Journal of Algorithms 39, 78–116 (2001)
Díaz, J., Petit, J., Serna, M.: Chapter titled “A Guide to Concentration Bounds. In: Handbook of Randomized Computing - Volumes I & II (Combinatorial Optimization 9), vol. I, pp. 457–507. Kluwer Academic Publishers, Dordrecht (2001)
Díaz, J., Petit, J., Serna, M.: A Random Graph Model for Optical Networks of Sensors. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, Springer, Heidelberg (2003); Also in the IEEE Transactions on Mobile Computing Journal 2(3),186–196 (2003)
Díaz, J., Petit, J., Serna, M.: Random Geometric Problems on [0, 1]2. In: Rolim, J.D.P., Serna, M., Luby, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 294–306. Springer, Heidelberg (1998)
Fill, J.A., Sheinerman, E.R., Singer-Cohen, K.B.: Random Intersection Graphs when m = ω(n): An Equivalence Theorem Relating the Evolution of the G(n, m, p) and G(n, p) models, http://citeseer.nj.nec.com/fill98random.html
Godehardt, E., Jaworski, J.: Two models of Random Intersection Graphs for Classification. In: Opitz, O., Schwaiger, M. (eds.) Studies in Classification, Data Analysis and Knowledge Organisation, pp. 67–82. Springer, Heidelberg (2002)
Karoński, M., Scheinerman, E.R., Singer- Cohen, K.B.: On Random Intersection Graphs: The Subgraph Problem. Combinatorics, Probability and Computing journal 8, 131–159 (1999)
Marczewski, E.: “Sur deux propriétés des classes d’ ensembles”. Fund. Math. 33, 303–307 (1945)
Nikoletseas, S., Raptopoulos, C., Spirakis, P.: The Existence and Efficient Construction of Large Independent Sets in General Random Intersection Graphs, http://www.cti.gr/RD1/nikole/english/psfiles/paper.ps
Penrose, M.: Random Geometric Graphs. Oxford Studies in Probability (2003)
Ross, S.M.: Stochastic Processes, 2nd edn. John Wiley & Sons, Chichester (1996)
Singer-Cohen, K.B.: Random Intersection Graphs, PhD thesis, John Hopkins University (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nikoletseas, S., Raptopoulos, C., Spirakis, P. (2004). The Existence and Efficient Construction of Large Independent Sets in General Random Intersection Graphs. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_86
Download citation
DOI: https://doi.org/10.1007/978-3-540-27836-8_86
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22849-3
Online ISBN: 978-3-540-27836-8
eBook Packages: Springer Book Archive