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 or 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. (ESA20) 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 a tight $(1-1/e-\epsilon)$ deterministic algorithm for SMKP. Our algorithm is based on reducing SMKP to an exponential-size submodular maximizaion problem over a special partition matroid which enjoys a tight deterministic algorithm. We develop several techniques to mimic the algorithm, leading to a tight deterministic approximation for SMKP.

翻译：子模块函数最大化是过去十年来计算机科学理论界的一个中心议题。 已经设计了大量运行良好的近效近似算法, 以便在各种制约下实现( mononoone 或非mononono) 子模块函数最大化。 在本文中, 我们考虑到子模块多宽差问题( SMKP), 这是一种研究周密的多重 knapsack 问题( MKP ) 的子模块版本。 粗略地说, 问题要求将单元子模块函数最大化于多个垃圾桶( knapsacks ) 。 最近, Fairstein et al. (ESA20) 为 SMKP 提出了一个紧凑( 1- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- imslon) 的子模块化算法。 其算法基于将紧凑 SMS- squal- squal 的算法, 我们的算算法将一个用于将一个稳定的固化的螺旋压到一个特殊的螺旋轴, 我们的螺旋压成一个特殊的螺旋压,, 将一个特殊的螺旋为一个特殊的螺旋至 的螺旋, 至一个特殊的螺旋。