Computing the independence number of intersection graphs

• Authors: Jacob Fox;János Pach
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NP-hard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n1/2+o(1)-approximation for α. Using extensions of the Lipton-Tarjan separator theorem, we improve this result and present, for every ε 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most nε. In contrast, for general graphs, for any ε 0 it is NP-hard to approximate the independence number within a factor of n1−ε. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane.