Bounds for online bin packing with cardinality constraints

We study a bin packing problem in which a bin can contain at most k   items of total size at most 1, where k≥2k≥2 is a given parameter. Items are presented one by one in an online fashion. We analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k≥4k≥4. Additionally, we present bounds for relatively small values of k   with respect to the asymptotic competitive ratio and the absolute competitive ratio. In particular, we provide tight bounds on the absolute competitive ratio of First Fit for k=2,3,4k=2,3,4, and improve the known lower bounds on asymptotic competitive ratios for multiple values of k. Our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general, and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing.