Improved Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint

Abstract

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 \((0.5-\varepsilon )\)-approximate solution in \(O(K\varepsilon ^{-1})\) 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 \((0.4-\varepsilon )\)-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and \(\varepsilon \). This improves on the previous best ratio of \(0.363-\varepsilon \) 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.