In this paper, we consider the problem of maximizing amonotone 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 - epsilon)-approximate solution in O( K epsilon(-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 thiswork, we propose a (0.4 - epsilon)-approximation algorithm for the knapsackconstrained problem, using space that is a polynomial of K and epsilon. This improves on the previous best ratio of 0.363 - epsilon 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.
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