The combinatorics of hidden diversity

In this paper, we study the following “balls in buckets” problem. Suppose there is a sequence B1,B2,…,Bn of buckets having integer sizes s1,s2,…,sn, respectively. For a given target fraction α, 0<α<1, our goal is to sequentially place balls in buckets until at least ⌈αn⌉ buckets are full, so as to minimize the number of balls used, which we shall denote by OPTα(I) for a given instance I.