Submodular maximization under various constraints is a fundamental problem studied continuously, in both computer science and operations research, since the late $1970$'s. A central technique in this field is to approximately optimize the multilinear extension of the submodular objective, and then round the solution. The use of this technique requires a solver able to approximately maximize multilinear extensions. Following a long line of work, Buchbinder and Feldman (2019) described such a solver guaranteeing $0.385$-approximation for down-closed constraints, while Oveis Gharan and Vondr\'ak (2011) showed that no solver can guarantee better than $0.478$-approximation. In this paper, we present a solver guaranteeing $0.401$-approximation, which significantly reduces the gap between the best known solver and the inapproximability result. The design and analysis of our solver are based on a novel bound that we prove for DR-submodular functions. This bound improves over a previous bound due to Feldman et al. (2011) that is used by essentially all state-of-the-art results for constrained maximization of general submodular/DR-submodular functions. Hence, we believe that our new bound is likely to find many additional applications in related problems, and to be a key component for further improvement.

翻译：子模最大化在各种约束下的研究自20世纪70年代末以来一直是计算机科学和运筹学领域的核心问题。该领域的关键技术是对子模目标函数的多线性扩展进行近似优化，随后对解进行舍入。应用这一技术要求求解器能够近似最大化多线性扩展。经过一系列长期研究工作，Buchbinder和Feldman（2019）提出了一种求解器，在向下闭约束下保证0.385近似比，而Oveis Gharan和Vondrák（2011）证明任何求解器无法实现优于0.478的近似比。本文提出一种达到0.401近似比的求解器，显著缩小了已知最优求解器与不可近似性结果之间的差距。该求解器的设计与分析基于我们为DR-submodular函数证明的新边界。该边界改进了Feldman等人（2011）的先前结果——该结果几乎被所有当前最优的一般子模/DR-submodular函数约束最大化方法所采用。因此，我们相信这一新边界将在相关问题的研究中获得更多应用，并成为推动进一步改进的关键组成部分。