An Asymptotic $\left(\frac{4}{3}+\varepsilon\right)$-Approximation for the 2-Dimensional Vector Bin Packing Problem

In this paper we consider the $2$-Dimensional Vector Bin Packing Problem (2VBP), a well-studied 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 two-dimensional 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 (from previous iterations) and samples a small number of configurations based on the solution for the configuration LP. For the analysis we derive an iteration-dependent upper bound on the solution size for the configuration LP, which holds with high probability. We also show that our Round&Round approach yields an AFPTAS for classic Bin Packing, suggesting its potential applicability for other variants of Bin Packing.