A note on maximum independent sets in rectangle intersection graphs

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 (1997) gave a (1+1/k)-factor algorithm with an O(n log n + n) time bound for any integer constant k ≥ 1; we describe a similar algorithm running in only O(n log n+ n∆) time, where ∆ ≤ n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan, and Ramaswami (2001) gave a dlogk ne-factor algorithm with an O(n ) time bound for any integer constant k ≥ 2; we describe similar algorithms running in O(n log n + n∆) and n log k) time.