Online Bin Packing with Overload Cost

Abstract

In the classical online bin packing problem, items arriving one by one with a given size not greater than 1 must be packed into unit-capacity bins such that the total size of items packed in a bin does not exceed its capacity; the objective is to minimize the total number of used bins. In this paper, we allow the total size of items packed in a bin to exceed the capacity, and there is a cost for each bin that depends on the total size of items assigned to it; in particular, overloading a bin, i.e., exceeding the capacity of a bin, comes at a prescribed cost. The corresponding goal is to minimize the total cost corresponding to the used bins. We pay 1 to open a bin with capacity 1, and we additionally pay c for each unit with which the bin is overloaded, i.e, the overload cost is linear in the size of the overload.

For each c, we present lower bounds on the competitive ratio achievable by deterministic algorithms. Further, we give an algorithm, called First-Fit Algorithm with Fixed Overload (FFO) that achieves the best possible competitive ratio for $c\le 3/2$. Furthermore, we sketch how the lower bounds apply to more general convex cost functions.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 754462 and funding from the NWO Gravitation Project NETWORKS, Grant Number 024.002.003.