Online bin packing with cardinality constraints resolved

TL;DR

This paper resolves a long-standing open problem in online bin packing with cardinality constraints by proving a lower bound of 2 on the asymptotic competitive ratio, matching the known upper bound and improving bounds for specific k values.

Contribution

It establishes a tight lower bound of 2 for the online bin packing with cardinality constraints, closing the open problem and introducing adaptive constructions for lower bounds.

Findings

Proves a lower bound of 2 on the competitive ratio for all k ≥ 2.

Improves lower bounds for specific values of k.

Introduces adaptive input constructions for lower bounds.

Abstract

Cardinality constrained bin packing or bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k \geq 2, items having sizes in (0,1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem in the sense that we prove a lower bound of 2 on the overall asymptotic competitive ratio. This closes this long standing open problem, since an algorithm of an absolute competitive ratio 2 is known. Additionally, we significantly improve the known lower bounds on the asymptotic competitive ratio for every specific value of k. The novelty of our constructions is based on full adaptivity that creates large gaps between item sizes. Thus, our lower bound inputs do not follow the common practice for online bin packing problems of having a known in advance input consisting of batches for which the algorithm needs to be competitive on every prefix of the input.