Tighter bounds for the harmonic bin packing algorithm

The harmonic algorithm, defined for online bin packing, partitions items into a fixed number M of classes of similar items, and packs each class independently and greedily in constant time for every packed item. The positive integer M is a parameter of the algorithm. This algorithm had a major role in the development of the online bin packing problem. Tight bounds on its asymptotic approximation ratio were known for M≤7, and for values of M with specific properties. The parametric variant of this algorithm, where item sizes are bounded from above by a certain value, was studied as well. We find tight bounds for many additional cases that were known as open, including the case M=8 for the classic problem.