Independent set of intersection graphs of convex objects in 2D

The intersection graph of a set of geometric objects is defined as a graph G (S, E) in which there is an edge between two nodes s(i), s(j) is an element of S if s(i) boolean AND s(j) not equal 0. The problem of computing a maximum independent set in the intersection graph of a set of objects is known to be NP-complete for most cases in two and higher dimensions. We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R-2. Specifically, given a set of n line segments in the plane with maximum independent set of size n, we present algorithms that find an independent set of size at least (i) (kappa/2 log(2n/kappa))(1/2) in time O(n(3)) and (ii) (kappa/2 log(2n/kappa))(1/4) in time O(n(4/3) log(c) n). For a set of n convex objects with maximum independent set of size kappa, we present an algorithm that finds an independent set of size at least (kappa/2 log(2n/kappa))(1/3) in time O(n(3) + tau(S)), assuming that S can be preprocessed in time tau(S) to answer certain primitive operations on these convex sets.