In [Math. Oper. Res., 2011], Fleischer et al. introduced a powerful technique for solving the generic class of separable assignment problems (SAP), in which a set of items of given values and weights needs to be packed into a set of bins subject to separable assignment constraints, so as to maximize the total value. The approach of Fleischer at al. relies on solving a configuration LP and sampling a configuration for each bin independently based on the LP solution. While there is a SAP variant for which this approach yields the best possible approximation ratio, for various special cases, there are discrepancies between the approximation ratios obtained using the above approach and the state-of-the-art approximations. This raises the following natural question: Can we do better by iteratively solving the configuration LP and sampling a few bins at a time? To assess the potential gain from iterative randomized rounding, we consider as a case study one interesting SAP variant, namely, Uniform Cardinality Constrained Multiple Knapsack, for which we answer this question affirmatively. The input is a set of items, each has a value and a weight, and a set of uniform capacity bins. The goal is to assign a subset of the items of maximum total value to the bins such that $(i)$ the capacity of any bin is not exceeded, and $(ii)$ the number of items assigned to each bin satisfies a given cardinality constraint. While the technique of Fleischer et al. yields a $\left(1-\frac{1}{e}\right)$-approximation for the problem, we show that iterative randomized rounding leads to an efficient polynomial time approximation scheme (EPTAS), thus essentially resolving the complexity status of the problem. Our analysis of iterative randomized rounding can be useful for solving other SAP variants.

翻译：在《数学运筹学》（2011年）中，Fleischer等人提出了一种强有力的技术，用于解决可分离分配问题（SAP）这一通用类别。该问题中，需要将一组具有给定价值和重量的物品装入一组受可分离分配约束的箱子中，以最大化总价值。Fleischer等人的方法依赖于求解配置线性规划，并基于该线性规划的解独立地对每个箱子进行配置采样。尽管对于SAP的某个变体，该方法能获得最优近似比，但在各种特殊情形下，使用上述方法获得的近似比与当前最优近似之间存在差距。这自然引出了以下问题：我们能否通过迭代求解配置线性规划并每次采样少数几个箱子来做得更好？为评估迭代随机舍入的潜在收益，我们以一个有趣的SAP变体——即均匀基数约束多背包问题——作为案例研究，并对此问题给出了肯定回答。输入包括一组物品（每个物品具有价值和重量）和一组容量均匀的箱子。目标是将总价值最大的物品子集分配给箱子，使得：(i) 任何箱子的容量不被超出，且 (ii) 分配给每个箱子的物品数量满足给定的基数约束。虽然Fleischer等人的技术能为该问题提供$\left(1-\frac{1}{e}\right)$-近似，但我们证明迭代随机舍入可导出高效多项式时间近似方案（EPTAS），从而基本解决了该问题的复杂度状态。我们对迭代随机舍入的分析有助于解决其他SAP变体。