A note on maximum independent sets in rectangle intersection graphs

Abstract

Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209–218] gave a (1+1/ k)-factor algorithm with an O( nlog n+ n 2 k−1 ) time bound for any integer constant k⩾1; we describe a similar algorithm running in only O( nlog n+ nΔ k−1 ) time, where Δ⩽ n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443–470] gave a ⌈log k n⌉-factor algorithm with an O( n k+1 ) time bound for any integer constant k⩾2; we describe similar algorithms running in O( nlog n+ nΔ k−2 ) and n O( k/log k) time.