Fully Dynamic Bin Packing

Abstract

Classic bin packing seeks to pack a given set of items of possibly varying sizes into a minimum number of identical sized bins. A number of approximation algorithms have been proposed for this NP-hard problem for both the on-line and off-line cases. In this chapter we discuss fully dynamic bin packing, where items may arrive (Insert) and depart (Delete) dynamically. In accordance with standard practice for fully dynamic algorithms, it is assumed that the packing may be arbitrarily rearranged to accommodate arriving and departing items. The goal is to maintain an approximately optimal solution of provably high quality in a total amount of time comparable to that used by an off-line algorithm delivering a solution of the same quality.

This chapter focuses on three results relative to fully dynamic bin packing. The first shows that imposing a fixed constant upper bound on the number of items that can be moved between bins per Insert/Delete operation forces the competitive ratio to be at least 4/3, regardless of the running time allowed per Insert/Delete. The second is a fully dynamic approximation algorithm for bin packing that is $\frac{5}{4}$ -competitive and that requires Θ(log n) time per Insert/Delete of an item. This competitive ratio of $\frac{5}{4}$ is nearly as good as that of the best practical off-line algorithms. A critical component of this algorithm is that very small items will be bundled together and moved as a single unit. Finally, we show for partially dynamic bin packing (Inserts only) and any ε>0, there is an algorithm with competitive ratio 1+ε that runs amortized polylogarithmic time.