(Probably) the minimum sum of squares

Abstract

We consider a combinatorial bin-packing problem known as minimum sum of squares: given a set of n piece sizes and a fixed number K of bins, pack the pieces so as to minimize the sum of the squares of the bin levels (hereafter called the cost). The problem is NP-complete in the strong sense. This note studies the distribution of the cost of the “block” packing rule, whose expected cost behavior was given by Nielsen [16]. Under the assumption that piece sizes are distributed uniformly on [0, γ], and given a desired degree of confidence 1 - e, if n piece sizes are sampled, then where is the cost using the block packing rule and is the cost of an optimum packing. Thus the performance of the rule is asymptotically optimal in probability, as well as in expectation.