A Tight Approximation for Fully Dynamic Bin Packing without Bundling

TL;DR

This paper presents a new algorithm for fully dynamic bin packing that maintains a near-optimal packing with minimal item repacking per step, improving upon previous methods by avoiding bundling of small items.

Contribution

The authors introduce a constant-repacking algorithm for fully dynamic bin packing that achieves a tight approximation ratio without relying on small item bundling.

Findings

Achieves an asymptotic approximation ratio of approximately 1.3871.

Repacks only a constant number of items per time step.

Closes the gap to the lower bound in the relaxed online bin packing problem.

Abstract

We consider a variant of the classical Bin Packing Problem, called Fully Dynamic Bin Packing. In this variant, items of a size in $(0,1]$ must be packed in bins of unit size. In each time step, an item either arrives or departs from the packing. An algorithm for this problem must maintain a feasible packing while only repacking a bounded number of items in each time step. We develop an algorithm which repacks only a constant number of items per time step and, unlike previous work, does not rely on bundling of small items which allowed those solutions to move an unbounded number of small items as one. Our algorithm has an asymptotic approximation ratio of roughly $1.3871$ which is complemented by a lower bound of Balogh et al., resulting in a tight approximation ratio for this problem. As a direct corollary, we also close the gap to the lower bound of the Relaxed Online Bin Packing Problem in which only insertions of items occur.