# Online Strip Packing with Polynomial Migration

## TL;DR

This paper introduces an online strip packing algorithm with polynomial amortized migration, achieving near-optimal packing ratios and allowing controlled repacking of items over time.

## Contribution

It presents the first online strip packing algorithm with a polynomial migration factor that attains an asymptotic ratio arbitrarily close to 1+epsilon.

## Key findings

- No constant migration factor algorithm beats a 4/3 ratio.
- The proposed AFPTAS achieves a ratio of 1+O(epsilon).
- Migration is polynomial in 1/epsilon.

## Abstract

We consider the relaxed online strip packing problem: Rectangular items arrive online and have to be packed without rotations into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i.e. to save migration for a later time step. As a main result, we present an AFPTAS with asymptotic ratio $1 + \mathcal{O}(\epsilon)$ for any $\epsilon > 0$ and amortized migration factor polynomial in $1 / \epsilon$. To our best knowledge, this is the first algorithm for online strip packing considered in a repacking model.