Algorithms for the Relaxed Online Bin-Packing Model

Abstract

The typical bin-packing problem requires the fitting of a sequence of rationals in (0,1] into a minimum number of bins of unit capacity, by packing the ith input element without any knowledge of the sizes or the number of input elements that follow. Moreover, unlike typical problems, this one issue does not admit any data reorganization, i.e., no element can be moved from one bin to another. In this paper, first of all, the Relaxed bin-packing model will be formalized; this model allows a constant number of elements to move from one bin to another, as a consequence of the arrival of a new input element. Then, in the context of this new model, two algorithms will be described. The first presents linear time and space complexities with a 1.5 approximation ratio and moves, at most once, only small elements; the second, instead, is an O(n log n) time and linear space algorithm with a 1.33. . . approximation ratio and moves each element a constant number of times. In the worst case, as a result of the arrival of a new input element, the first algorithm moves no than three elements, while the second moves as many as seven elements. Please note that the number of movements performed is explicitly considered in the complexity analysis. Both algorithms are below the theoretical 1.536. . . lower bound, effective for the bin-packing algorithms without the movement of elements. Moreover, our algorithms are more online than any other linear space bin-packing algorithm because, unlike the algorithms already known, they allow the return of a (possibly relevant) fraction of bins before the work is carried out.