In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a $$(0.5-\varepsilon )$$ ( 0.5 - ε ) -approximate solution in $$O(K\varepsilon ^{-1})$$ O ( K ε - 1 ) space, where K is the knapsack capacity (Badanidiyuru et al. KDD 2014). The approximation ratio is recently shown to be optimal (Feldman et al. STOC 2020). In this work, we propose a $$(0.4-\varepsilon )$$ ( 0.4 - ε ) -approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and $$\varepsilon $$ ε . This improves on the previous best ratio of $$0.363-\varepsilon $$ 0.363 - ε with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one.

中文翻译：

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在背包约束下最大化单调子模函数的改进流算法

在本文中，我们考虑了在流设置中最大化受背包约束的单调子模函数的问题。在这样的设置中，元素在任何时间点按顺序到达，并且算法只能存储到目前为止已经到达的元素的一小部分。对于所有元素都有单位大小的特殊情况（即基数约束情况），可以在 $$O(K\varepsilon )$$ ( 0.5 - ε ) - 近似解^{-1})$$ O ( K ε - 1 ) 空间，其中 K 是背包容量（Badanidiyuru 等人，KDD 2014）。最近证明近似比率是最佳的（Feldman 等人，STOC 2020）。在这项工作中，我们为背包约束问题提出了一个 $$(0.4-\varepsilon )$$ ( 0.4 - ε ) -近似算法，使用空间是 K 和 $$\varepsilon $$ ε 的多项式。这改进了之前的最佳比率 $0.363-\varepsilon $$0.363 - ε，空间相同。我们的算法基于各种想法的仔细组合，将多通道流算法转换为单通道算法。