We study the Uniform Cardinality Constrained Multiple Knapsack problem (CMK), a natural generalization of Multiple Knapsack with applications ranging from cloud computing to radio networks. 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. The best known approximation ratio for CMK is $1-\frac{\ln (2)}{2} -\epsilon \approx 0.653$, which follows from a result for a generalization of the problem. Our main contribution is an efficient polynomial time approximation scheme (EPTAS) for CMK. This essentially resolves the complexity status of the problem, since the existence of a fully polynomial time approximation scheme (FPTAS) is ruled out. Our technique is based on the following simple algorithm: in each iteration, solve a configuration linear program (LP) of the problem; then, sample configurations (i.e., feasible subsets of items for a single bin) according to a distribution specified by the LP solution. The algorithm terminates once each bin is assigned a configuration. We believe that our generic technique may lead to efficient approximations for other assignment problems.

翻译：我们研究均匀基数约束多重背包问题（CMK），这是多重背包问题的自然推广，应用领域涵盖云计算到无线网络。输入包含一组物品（每个具有价值和重量）以及一组容量统一的箱子。目标是将物品子集分配至箱子，使得总价值最大化，同时满足：(i) 任何箱子容量不被超出，(ii) 每个箱子分配的物品数量满足给定基数约束。CMK已知的最佳近似比为$1-\frac{\ln (2)}{2} -\epsilon \approx 0.653$，该结果源于对问题推广形式的研究。我们的主要贡献是为CMK提出了一个高效多项式时间近似方案（EPTAS）。这实质上解决了问题的复杂度状态，因为完全多项式时间近似方案（FPTAS）的存在性已被排除。我们的技术基于以下简单算法：每次迭代中，求解问题的配置线性规划（LP）；然后根据LP解指定的分布采样配置（即单个箱子的可行物品子集）。算法在每只箱子分配一个配置后终止。我们相信这种通用技术可能为其他指派问题带来高效近似算法。