Independent Set of Intersection Graphs of Convex Objects in 2D

Abstract

AbstractThe 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 ∈ S if s i ∩ s j ≠ ∅. 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 ℝ2. Specifically, given a set of n line segments in the plane with maximum independent set of size κ, we present algorithms that find an independent set of size at least (i) (κ/2log (2n/κ))1/2 in time O(n 3) and (ii) (κ/2log (2n/κ))1/4 in time O(n 4/3 logc n). For a set of n convex objects with maximum independent set of size κ, we present an algorithm that finds an independent set of size at least (κ/2log (2n/κ) )1/3 in time O(n 3 + τ(S)), assuming that S can be preprocessed in time τ(S) to answer certain primitive operations on these convex sets.KeywordsApproximation RatioGeometric ObjectIntersection GraphGeometric GraphConvex ObjectThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.