Constructions of independent sets in random intersection graphs

This paper concerns constructing independent sets in a random intersection graph. We concentrate on two cases of the model: a binomial and a uniform random intersection graph. For both models we analyse two greedy algorithms and prove that they find asymptotically almost optimal independent sets. We provide detailed analysis of the presented algorithms and give tight bounds on the independence number for the studied models. Moreover we determine the range of parameters for which greedy algorithms give better results for a random intersection graph than this is in the case of an Erd?s-Renyi random graph G ( n , p ? ) .

[1]  Mindaugas Bloznelis.  Degree distribution of a typical vertex in a general random intersection graph , 2008 .

[2]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[3]  Alan M. Frieze,et al.  On the independence number of random graphs , 1990, Discret. Math..

[4]  Paul G. Spirakis,et al.  On the independence number and Hamiltonicity of uniform random intersection graphs , 2011, Theor. Comput. Sci..

[5]  James Allen Fill,et al.  Random intersection graphs when m= w (n): an equivalence theorem relating the evolution of the G ( n, m, p ) and G ( n,P /italic>) models , 2000 .

[6]  Andreas N. Lagerås,et al.  A Note on the Component Structure in Random Intersection Graphs with Tunable Clustering , 2008, Electron. J. Comb..

[7]  Duncan J. Watts,et al.  The Structure and Dynamics of Networks: (Princeton Studies in Complexity) , 2006 .

[8]  Roberto Di Pietro,et al.  Sensor Networks that Are Provably Resilient , 2006, 2006 Securecomm and Workshops.

[9]  Mindaugas Bloznelis,et al.  Component evolution in a secure wireless sensor network , 2009 .

[10]  Katarzyna Rybarczyk,et al.  The Degree Distribution in Random Intersection Graphs , 2010, GfKl.

[11]  B. Bollobás.  The evolution of random graphs , 1984 .

[12]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[13]  Edward R. Scheinerman,et al.  On Random Intersection Graphs: The Subgraph Problem , 1999, Combinatorics, Probability and Computing.

[14]  Paul G. Spirakis,et al.  Large independent sets in general random intersection graphs , 2008, Theor. Comput. Sci..

[15]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[16]  Katarzyna Rybarczyk,et al.  Diameter, connectivity, and phase transition of the uniform random intersection graph , 2011, Discret. Math..

[17]  Michael Behrisch,et al.  Component Evolution in Random Intersection Graphs , 2007, Electron. J. Comb..

[18]  Maurizio Vichi,et al.  Studies in Classification Data Analysis and knowledge Organization , 2011 .

[19]  Colin McDiarmid,et al.  Algorithmic theory of random graphs , 1997 .

[20]  Erhard Godehardt,et al.  Two Models of Random Intersection Graphs for Classification , 2003 .

[21]  Paul Erdös,et al.  On random graphs, I , 1959 .

[22]  Mindaugas Bloznelis,et al.  Degree and clustering coefficient in sparse random intersection graphs , 2013, 1303.3388.

[23]  Katarzyna Rybarczyk,et al.  Equivalence of a random intersection graph and G(n,p) , 2009, Random Struct. Algorithms.

[24]  B. Bollobás,et al.  Cliques in random graphs , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[25]  Willemien Kets,et al.  RANDOM INTERSECTION GRAPHS WITH TUNABLE DEGREE DISTRIBUTION AND CLUSTERING , 2009, Probability in the Engineering and Informational Sciences.