Packing random rectangles

Abstract

A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0, 1]. We prove that the number Cn of items in a maximum cardinality disjoint subset of n random rectangles satisfies n1/2/K ≤ ECn ≤ Kn1/2, where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constant K, n1/2/K ≤ ECn ≤ K (n logd-1 n)1/2. Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Qn of items in a maximum cardinality disjoint subset of the cubes satisfies nd/(d+1)/K ≤ EQn ≤ Knd/(d+1).