An Asymptotic $\left(\frac{4}{3}+\varepsilon\right)$Approximation for the 2Dimensional Vector Bin Packing Problem
Abstract
We study the $2$Dimensional Vector Bin Packing Problem (2VBP), a generalization of classic Bin Packing that is widely applicable in resource allocation and scheduling. In 2VBP we are given a set of items, where each item is associated with a twodimensional volume vector. The objective is to partition the items into a minimal number of subsets (bins), such that the total volume of items in each subset is at most $1$ in each dimension. We give an asymptotic $\left(\frac{4}{3}+\varepsilon\right)$approximation for the problem, thus improving upon the best known asymptotic ratio of $\left(1+\ln \frac{3}{2}+\varepsilon\right)\approx 1.406$ due to Bansal, Elias and Khan (SODA 2016). Our algorithm applies a novel Round&Round approach which iteratively solves a configuration LP relaxation for the residual instance and samples a small number of configurations based on the solution for the configuration LP. For the analysis we derive an iterationdependent upper bound on the solution size for the configuration LP, which holds with high probability. To facilitate the analysis, we introduce key structural properties of 2VBP instances, leveraging the recent fractional grouping technique of Fairstein et al. (ESA 2021).
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.12828
 Bibcode:
 2022arXiv220512828K
 Keywords:

 Computer Science  Data Structures and Algorithms