We study the uniform $2$-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a $2$-dimensional weight vector and a positive profit, along with $m$ $2$-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized. Our main result is a $(1- \frac{\ln 2}{2} - \varepsilon)$-approximation algorithm for 2VMK, for every fixed $\varepsilon > 0$, thus improving the best known ratio of $(1 - \frac{1}{e}-\varepsilon)$ which follows as a special case from a result of [Fleischer at al., MOR 2011]. Our algorithm relies on an adaptation of the Round$\&$Approx framework of [Bansal et al., SICOMP 2010], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to $\approx m\cdot \ln 2 \approx 0.693\cdot m$ of the bins, followed by a reduction to the ($1$-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.

翻译：我们研究均匀二维向量多背包（2VMK）问题，这是多背包问题的一种自然变体，在虚拟机部署等实际应用中具有重要价值。2VMK问题的输入包含一组物品，每个物品具有二维权重向量和正利润，以及m个在每一维度上具有均匀（单位）容量的二维箱子。目标是将物品子集分配至各箱子，使得每个箱子在各维度上承载物品的总重量不超过一，同时最大化总利润。我们的主要成果是为2VMK设计了一个针对任意固定ε>0的(1 - ln2/2 - ε)近似算法，从而改进了由[Fleischer等, MOR 2011]的结果作为特例推导出的已知最佳比率(1 - 1/e - ε)。该算法借鉴了[Bansal等, SICOMP 2010]最初为集合覆盖问题设计的Round&Approx框架，将其适配至最大化问题。算法通过对配置线性规划的解进行随机舍入，将物品分配至约m·ln2 ≈ 0.693·m个箱子，随后将剩余箱子的物品分配问题简化为（一维）多背包问题。