Random-order bin packing

An instance of the classical bin packing problem consists of a positive real C and a list L = (a1, a2, ..., an) of items with sizes 0 < s(ai) ≤ C, 1 ≤ i ≤ n; a solution to the problem is a partition of L into a minimum number of blocks, called bins, such that the sum of the sizes of the items in each bin is at most the capacity C. The capacity is just a scaling parameter; as is customary, we put C = 1, and restrict item sizes to the unit interval. Research on the bin packing problem started over 30 years ago [GGU72], [Joh73]. As the problem is NP-complete [GJ79], many approximation algorithms have been proposed and analyzed. Next Fit (NF) is arguably the most elementary, as it packs items bin by bin, not starting a new bin until an item is encountered that does not fit into the current, open bin; in this event the open bin is closed, the new bin becomes the open bin, and no further attempt is made to pack items in the bin just closed. A natural generalization of NF is the First Fit algorithm (FF), which never closes bins; it packs each successive item from L in the first (lowest indexed) bin which has enough space for it. Another improvement on NF is the Best Fit algorithm (BF), which packs the next item in the lowest indexed bin, among those in which it fits, which has the maximum sum of item sizes. The most common ways of appraising an approximation algorithm are performance ratios, which give the performance of an approximation algorithm relative to an optimal algorithm. Informally, asymptotic bounds for algorithm A typically take the form: For given constants α ≥ 1, β ≥ 0, A(L) ≤ αOPT (L)+β