Submodular function maximization has been a central topic in the theoretical computer science community over the last decade. Plenty of well-performing approximation algorithms have been designed for the maximization of monotone/non-monotone submodular functions over a variety of constraints. In this paper, we consider the submodular multiple knapsack problem (SMKP), which is the submodular version of the well-studied multiple knapsack problem (MKP). Roughly speaking, the problem asks to maximize a monotone submodular function over multiple bins (knapsacks). Recently, Fairstein et al. [13] presented a tight $(1-1/e-\epsilon)$-approximation randomized algorithm for SMKP. Their algorithm is based on the continuous greedy technique which inherently involves randomness. However, the deterministic algorithm of this problem has not been understood very well previously. In this paper, we present deterministic algorithms with improved approximation ratios for SMKP. We first consider the case when the number of bins is a constant and provide a simple combinatorial deterministic algorithm with an optimal $(1-1/e)$ ratio. Previously, a randomized approximation algorithm obtained a $(1 - 1/e-\epsilon)$ approximation ratio based on the involved continuous greedy technique. We then generalize the result to arbitrary number of bins. When the capacity of bins are identical, we design a combinatorial and deterministic algorithm which achieves an almost tight approximation ratio $(1 - 1 / e-\epsilon)$. In the general case, we provide a $(1/2-\epsilon)$-approximation algorithm which is also combinatorial and deterministic. We finally boost this algorithm to a $(1-1/e-\epsilon)$ randomized algorithm for the general case, thus matching the result of Fairstein et al. [13].

中文翻译：

---

子模多背包问题的确定性算法

在过去十年中，子模块函数最大化一直是理论计算机科学界的中心话题。已经设计了大量性能良好的近似算法，用于在各种约束条件下最大化单调/非单调子模函数。在本文中，我们考虑了子模块多背包问题 (SMKP)，它是经过充分研究的多背包问题 (MKP) 的子模块版本。粗略地说，该问题要求在多个箱（背包）上最大化单调子模函数。最近，费尔斯坦等人。[13] 为 SMKP 提出了一个紧密的 $(1-1/e-\epsilon)$-近似随机算法。他们的算法基于固有地涉及随机性的连续贪婪技术。然而，这个问题的确定性算法以前不是很了解。在本文中，我们提出了具有改进的 SMKP 逼近比的确定性算法。我们首先考虑 bin 数量为常数的情况，并提供具有最佳 $(1-1/e)$ 比率的简单组合确定性算法。以前，随机逼近算法基于所涉及的连续贪婪技术获得了 $(1 - 1/e-\epsilon)$ 逼近比。然后我们将结果推广到任意数量的 bin。当 bins 的容量相同时，我们设计了一个组合和确定性算法，该算法实现了几乎严格的逼近比 $(1 - 1 / e-\epsilon)$。在一般情况下，我们提供了一个 $(1/2-\epsilon)$-近似算法，它也是组合性和确定性的。对于一般情况，我们最终将此算法提升为 $(1-1/e-\epsilon)$ 随机算法，从而匹配 Fairstein 等人的结果。[13]。