Large independent sets in general random intersection graphs

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 , p → ) which includes the model of [M. Karoński, E.R. Scheinerman, K.B. Singer-Cohen, On random intersection graphs: The subgraph problem, Combinatorics, Probability and Computing journal 8 (1999), 131–159] (the “uniform” random intersection graph models) 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.