Online Bin Packing: Old Algorithms and New Results

In the bin packing problem we are given an instance consisting of a sequence of items with sizes between 0 and 1. The objective is to pack these items into the smallest possible number of bins of unit size. FirstFit and BestFit algorithms are simple online algorithms introduced in early seventies, when it was also shown that their asymptotic approximation ratio is equal to 1.7. We present a simple proof of this bound and survey recent developments that lead to the proof that also the absolute approximation ratio of these algorithms is exactly 1.7. More precisely, if the optimum needs opt bins, the algorithms use at most \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins and for each value of opt, there are instances that actually need so many bins. We also discuss bounded-space bin packing, where the online algorithm is allowed to keep only a fixed number of bins open for future items. In this model, a variant of BestFit also has asymptotic approximation ratio 1.7, although it is possible that the bound is significantly smaller if also the offline solution is required to satisfy the bounded-space restriction.