# Independent set of intersection graphs of convex objects in 2D

Published

Journal Article

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≠ ∅. 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 α, we present algorithms that find an independent set of size at least (α/(2log(2n/α))) 1/2 in time O(n 3) and (α/(2log(2n/α))) 1/4 in time O(n 4/3log cn), (ii) a set of n convex objects with maximum independent set of size α, we present an algorithm that finds an independent set of size at least (α/(2log(2n/α))) 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, and (iii) a set of n rectangles with maximum independent set of size βn, for β≤1, we present an algorithm that computes an independent set of size Ω( β 2n) . All our algorithms use the notion of partial orders that exploit the geometric structure of the convex objects. © 2006 Elsevier B.V. All rights reserved.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Mustafa, NH

### Published Date

- May 1, 2006

### Published In

### Volume / Issue

- 34 / 2

### Start / End Page

- 83 - 95

### International Standard Serial Number (ISSN)

- 0925-7721

### Digital Object Identifier (DOI)

- 10.1016/j.comgeo.2005.12.001

### Citation Source

- Scopus