Independent set of intersection graphs of convex objects in 2D

  • Authors:

  • Pankaj K. Agarwal;Nabil H. Mustafa

  • Affiliations:

  • Department of Computer Science, Duke University, Durham, NC 27708-0129, USA;Department of Computer Science, Duke University, Durham, NC 27708-0129, USA

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2006

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Abstract

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@?S if s"i@?s"j@A. 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 (i) a set of n line segments in the plane with maximum independent set of size @a, we present algorithms that find an independent set of size at least (@a/(2log(2n/@a)))^1^/^2 in time O(n^3) and (@a/(2log(2n/@a)))^1^/^4 in time O(n^4^/^3log^cn), (ii) a set of n convex objects with maximum independent set of size @a, we present an algorithm that finds an independent set of size at least (@a/(2log(2n/@a)))^1^/^3 in time O(n^3+@t(S)), assuming that S can be preprocessed in time @t(S) to answer certain primitive operations on these convex sets, and (iii) a set of n rectangles with maximum independent set of size @bn, for @b=