We study the $d$-dimensional Vector Bin Packing ($d$VBP) problem, a generalization of Bin Packing with central applications in resource allocation and scheduling. In $d$VBP, we are given a set of items, each of which is characterized by a $d$-dimensional volume vector; the objective is to partition the items into a minimum number of subsets (bins), such that the total volume of items in each subset is at most $1$ in each dimension. Our main result is an asymptotic approximation algorithm for $d$VBP that yields a ratio of $(1+\ln d-\chi(d) +\varepsilon)$ for all $d \in \mathbb{N}$ and any $\varepsilon > 0$; here, $\chi(d)$ is some strictly positive function. This improves upon the best known asymptotic ratio of $ \left(1+ \ln d +\varepsilon\right)$ due to Bansal, Caprara and Sviridenko (SICOMP 2010) for any $d >3$. By slightly modifying our algorithm to include an initial matching phase and applying a tighter analysis we obtain an asymptotic approximation ratio of $\left(\frac{4}{3}+\varepsilon\right)$ for the special case of $d=2$, thus substantially improving the previous best ratio of $\left(\frac{3}{2}+\varepsilon\right)$ due to Bansal, Elias and Khan (SODA 2016). Our algorithm 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. While iterative rounding was already used by Karmarkar and Karp (FOCS 1982) to establish their celebrated result for classic (one-dimensional) Bin Packing, iterative randomized rounding is used here for the first time in the context of (Vector) Bin Packing. Our results show that iterative randomized rounding is a powerful tool for approximating $d$VBP, leading to simple algorithms with improved approximation guarantees.

翻译：我们研究 $d$ 维向量装箱问题（$d$VBP），这是装箱问题的一种推广，在资源分配与调度领域具有核心应用。在 $d$VBP 中，给定一组物品，每个物品具有一个 $d$ 维体积向量；目标是将物品划分为尽可能少的子集（箱子），使得每个子集中物品在每一维度上的总体积不超过 $1$。我们的主要结果为 $d$VBP 设计了一种渐近近似算法，对任意 $d \in \mathbb{N}$ 和 $\varepsilon > 0$，该算法达到 $(1+\ln d-\chi(d) +\varepsilon)$ 的近似比，其中 $\chi(d)$ 为某严格正函数。这改进了 Bansal、Caprara 和 Sviridenko (SICOMP 2010) 针对所有 $d>3$ 情形建立的最佳已知渐近比 $(1+\ln d+\varepsilon)$。通过轻微修改算法引入初始匹配阶段并采用更紧致的分析，我们在 $d=2$ 的特殊情形下获得 $\left(\frac{4}{3}+\varepsilon\right)$ 的渐近近似比，从而大幅改进了 Bansal、Elias 和 Khan (SODA 2016) 此前的最佳结果 $\left(\frac{3}{2}+\varepsilon\right)$。我们的算法迭代求解残余实例（来自先前迭代）的配置 LP 松弛，并基于配置 LP 的解采样少量配置。尽管 Karmarkar 和 Karp (FOCS 1982) 曾使用迭代舍入法建立经典（一维）装箱问题的著名结果，但本文首次在（向量）装箱问题中应用迭代随机舍入。研究结果表明，迭代随机舍入是逼近 $d$VBP 问题的有力工具，能够导出具有改进近似保证的简洁算法。