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 ≠ ∅. 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.
Work has been supported by NSF under grants CCR-00-86013 EIA-98-70724, EIA-99-72879, EIA-01-31905, and CCR-02-04118 and by a grant from the U.S.-Israeli Binational Science Foundation.
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Agarwal, P.K., Mustafa, N.H. (2004). Independent Set of Intersection Graphs of Convex Objects in 2D. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_12
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DOI: https://doi.org/10.1007/978-3-540-27810-8_12
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