Submodular maximization has been a central topic in theoretical computer science and combinatorial optimization over the last decades. Plenty of well-performed approximation algorithms have been designed for the problem over a variety of constraints. In this paper, we consider the submodular multiple knapsack problem (SMKP). In SMKP, the profits of each subset of elements are specified by a monotone submodular function. The goal is to find a feasible packing of elements over multiple bins (knapsacks) to maximize the profit. Recently, Fairstein et al.~[ESA20] proposed a nearly optimal $(1-e^{-1}-\epsilon)$-approximation algorithm for SMKP. Their algorithm is obtained by combining configuration LP, a grouping technique for bin packing, and the continuous greedy algorithm for submodular maximization. As a result, the algorithm is somewhat sophisticated and inherently randomized. In this paper, we present an arguably simple deterministic combinatorial algorithm for SMKP, which achieves a $(1-e^{-1}-\epsilon)$-approximation ratio. Our algorithm is based on very different ideas compared with Fairstein et al.~[ESA20].

翻译：次模最大化在过去几十年中一直是理论计算机科学与组合优化的核心课题。针对各种约束下的该问题，研究者已设计出大量性能优异的近似算法。本文考虑次模多背包问题（SMKP）。在该问题中，每个元素子集的收益由单调次模函数定义，目标是在多个背包（箱子）中寻找可行的元素打包方案以最大化收益。近期，Fairstein 等人 [ESA20] 提出了 SMKP 近乎最优的 $(1-e^{-1}-\epsilon)$-近似算法。该算法通过结合配置线性规划、装箱分组技术以及次模最大化的连续贪心算法实现，因此较为复杂且本质上是随机化的。本文提出一种可论证简洁的确定性组合算法，该算法同样达到 $(1-e^{-1}-\epsilon)$-近似比，其核心思路与 Fairstein 等人 [ESA20] 的方法截然不同。