Independent Sets in Random Intersection Graphs

Years and Authors of Summarized Original Work

• 2004; Nikoletseas, Raptopoulos, Spirakis

Problem Definition

This problem is concerned with the efficient construction of an independent set of vertices (i.e., a set of vertices with no edges between them) with maximum cardinality, when the input is an instance of the uniform random intersection graphs model. This model was introduced by Karoński, Sheinerman, and Singer-Cohen in [4] and Singer-Cohen in [10] and it is defined as follows

Definition 1 (Uniform random intersection graph)

Consider a universe ${ M {=} \{1, 2, \dots, m\} }$ of elements and a set of vertices ${ V {=} \{ v_1, v_2, \dots, v_n\!\} }$. If one assigns independently to each vertex ${ v_j, j = 1, 2, \dots, n }$, a subset ${ S_{v_j} }$ of M by choosing each element independently with probability p and puts an edge between two vertices ${ v_{j_1}, v_{j_2} }$ if and only if ${ S_{v_{j_1}} \cap S_{v_{j_2}} \neq \emptyset }$, then the resulting graph is...