We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (Math Oper Res 42(4):1197–1218, 2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of Sviridenko et al. (2017). We also show that the guarantee of our algorithm becomes slightly better when the polytope is down-monotone, and that this better guarantee is tight for such polytopes.

中文翻译：

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猜测子模和线性和的自由最大化

我们考虑最大化单调子模函数和受一般可解多面体约束的线性函数之和的问题。最近，Sviridenko 等人。(Math Oper Res 42(4):1197–1218, 2017) 描述了针对这个问题的算法，其近似保证在某些直观和形式意义上是最优的。不幸的是，这个算法涉及一个猜测步骤，这使得它不太干净并且显着影响了它的时间复杂度。在这项工作中，我们描述了一种干净的替代算法，它使用一种新颖的加权技术来避免有问题的猜测步骤，同时保持与 Sviridenko 等人的算法相同的近似保证。(2017)。我们还表明，当多胞体是向下单调时，我们算法的保证会稍微好一点，