The tight asymptotic approximation ratio of First Fit for bin packing with cardinality constraints

In bin packing with cardinality constraints (BPCC), there is an upper bound k≥2 on the number of items that can be packed into each bin, additionally to the standard constraint on the total size of items. We study the algorithm First Fit (FF), acting on a list of items, packing each item into the minimum indexed bin that contains at most k−1 items and has sufficient space for the item. We present a complete analysis of its asymptotic approximation ratio for all values of k. Many years after FF for BPCC was introduced, its tight asymptotic approximation ratio is finally found.